# Univalent Logharmonic Mappings in the Plane

Authors: Zayid Abdulhadi; Rosihan M. Ali
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper surveys recent advances on univalent logharmonic mappings defined on a simply or multiply connected domain. Topics discussed include mapping theorems, logharmonic automorphisms, univalent logharmonic extensions onto the unit disc or the annulus, univalent logharmonic exterior mappings, and univalent logharmonic ring mappings. Logharmonic polynomials are also discussed, along with several important subclasses of logharmonic mappings.


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## Body

## 1. Introduction

LetD be a domain in the complex plane ℂ. Denote by H(D) (resp., by M(D)) the linear space of all analytic (resp., meromorphic) functions in D, and let B(D) be the set of all functions a∈H(D) satisfying |a(z)|<1, z∈D. A nonconstant function f is logharmonic in D if f is the solution of the nonlinear elliptic differential equation(1.1)fz¯¯=af¯ffz,a∈B(D). The function a is called the second dilatation of f. In contrast to the linear space H(D) consisting of analytic functions, translations in the image do not preserve logharmonicity, and the inverse of a logharmonic function is not necessarily logharmonic. If f1 and f2 are two logharmonic functions with respect to a∈B(D), then f1·f2 is logharmonic with respect to the same a. If, in addition, 0∉f2(D), then f1/f2 is also logharmonic. The composition f∘ϕ of a logharmonic mapping f with a conformal premapping ϕ is also logharmonic with respect to a∘ϕ. However, the composition ϕ∘f of a conformal postmapping ϕ with a logharmonic mapping f is in general not logharmonic. If f is a logharmonic mapping in D, then f is a nonconstant locally quasiregular mapping, and, therefore, it is continuous, open, and light. It follows that f can be represented as a composition of two functions f=A∘χ, where χ is a locally quasiconformal homeomorphism in D and A∈H(χ(D)). As an immediate consequence, the maximum principle, the identity principle, and the argument principle all still hold for logharmonic mappings.The study of logharmonic mappings was initiated in the main by Abdulhadi, Bshouty, and Hengartner in the last century, and the basic theory of logharmonic mappings was developed in [1–8].A local representation for logharmonic mappings was given by Abdulhadi and Bshouty in [1]. In particular, they obtained the following result.Theorem 1.1.
Letf be a logharmonic mapping in D with respect to a∈B(D). Suppose that f(z0)=0 and B(z0,ρ)∖{z0}⊂D∖Z(f), where B(z0,ρ)={z:|z-z0|<ρ} and Z(f)={z∈D:f(z)=0}. Then f admits the representation
(1.2)f(z)=(z-z0)|z-z0|2βnh(z)g(z)̅,z∈B(z0,ρ),
where n∈ℕ, β=na(z0)¯(1+a(z0))/(1-|a(z0)|2) and, therefore, Re(β)>-n/2. The functions h and g are in H(B(z0,ρ)), with h(z0)≠0 and g(z0)=1.As a direct consequence of Theorem1.1, we have the following global representation for logharmonic mappings.Corollary 1.2.
LetD be a simply connected domain in ℂ and f a logharmonic mapping in D. If f has exactly p zeros {zk}k=1p in D   (counting multiplicities), then f admits a global representation given by
(1.3)f(z)=[∏k=1p(z-zk)|z-zk|2βk]h(z)g(z)¯,
where βk=a(zk)¯(1+a(zk))/(1-|a(zk)|2) and, therefore, Re(βk)>-1/2. The functions h and g are in H(D), and 0∉h·g(D).For the converse, Abdulhadi and Hengartner [2] proved the following theorem.Theorem 1.3.
Suppose thatf(z)=h(z)g(z)¯ is defined in a domain D, where h and g are in H(D), such that f(D) does not lie on a logarithmic spiral. Then either f=g¯ or f is a solution of
(1.4)fz¯(z)¯=af(z)¯f(z)fz(z),a∈M(D),|a|≠1.Remark 1.4.
The converse of Theorem1.3 does not hold. Indeed, consider the partial differential equation fz¯¯=(1/3)(f¯/f)fz. Then f1(z)=z6z¯2 and f2(z)=z|z| are solutions of this equation. The function f1 can be written in the form hg¯ while f2 could not.Remark 1.5.
The functiongw(z)=f(z)-w,w∈ℂ, cannot be written in the form hg¯ unless w=0 or f is a constant. However, it is a solution of the second Beltrami equation
(1.5)(∂gw(z)∂z¯)¯=μw(z,gw)∂gw(z)∂z,
where
(1.6)μw(z,gw)=a(z)gw(z)+w¯gw(z)+w.
Hence, |μw|≡|a| in D and is independent of w.Corollary 1.6.
The imagef(D) of a nonconstant function f(z)=h(z)g(z)¯ lies on a logarithmic spiral if and only if f is a solution of (1.1) with |a|≡1.In the theory of quasiconformal mappings, it is proved that, for any measurable functionμ with |μ|<1, the solution of Beltrami equation fz¯=μfz can be factorized in the form f=ψ∘F, where F is a univalent quasiconformal mapping and ψ is an analytic function (see [9]). For sense-preserving harmonic mappings, the answer is negative. In [10], Duren and Hengartner gave a necessary and sufficient condition on sense-preserving harmonic mappings f for the existence of such a factorization. Moreover, for logharmonic mappings, such a factorization need not exist. For example, the function f(z)=z2/|1-z|4 is a sense-preserving logharmonic mapping with respect to a(z)=z, and f has no decomposition of the desired form (see [11]). The following factorization theorem was proved in [11].Theorem 1.7.
Letf be a nonconstant logharmonic mapping defined in a domain D⊂ℂ, and let a be its second dilatation function. Then f can be factorized in the form f=F∘φ, for some analytic function φ and some univalent logharmonic mapping F if and only if(a)
|a(z)|≠1 in D,(b)
f(z1)=f(z2) implies a(z1)=a(z2).Under these conditions, the representation is unique up to a conformal mapping; any other representationf=F1∘φ1 has the form F1=F∘ψ-1 and φ1=ψ∘φ for some conformal mapping defined in φ(D).Consider now the logharmonic mappingf(z)=ze1/ze-1/z¯. The point z=0 is an isolated singularity of f, and f is continuous at this point. However, f does not admit a logharmonic-continuation to ℂ. A further restriction is needed.Theorem 1.8 (see [2] (logharmonic-continuation across an isolated singularity)).
LetDr be the point disc Dr={z:0<|z|<r}, and let f=hg¯ defined in Dr be a logharmonic mapping with respect to a∈B(D) satisfying limz→0f(z)=0. Then f admits a logharmonic-continuation across the origin and has the representation
(1.7)f(z)=zn0z¯m0h0(z)g0(z)¯,
where n0 and m0 are nonnegative integers, 0≤m0<n0, and h0 and g0 are analytic functions on |z|<r satisfying h0(0)g0(0)≠0.Liouville’s theorem does not hold for entire logharmonic functions. The functionf(z)=exp(z)exp(-z¯) is a nonconstant bounded logharmonic in ℂ. Its dilatation is a(z)≡-1. However, the following modified version of Liouville’s theorem was given in [2].Theorem 1.9 (modified Liouville’s theorem).
Letf=hg¯ be a bounded logharmonic function in ℂ. Then either the image  f(ℂ) is a circle centered at the origin with dilatation function a(z)≡-1 or f is a constant.Letf(z)=h(z)g(z)¯ be a logharmonic mapping defined in a domain D with respect to a∈B(D) satisfying |a(z)|≢1. Let(1)
SG(D)={z∈D:|a(z)|>1},(2)
SL(D)={z∈D:|a(z)|<1},(3)
SE(D)={z∈D:|a(z)|=1},(4)
NZ(f-w,D) be the cardinality of Z(f-w,D), that is, the number of zeros of f-w in D, multiplicity is not counted,(5)
VZ(f-w,G) be the number of zeros of f-w in SG(D), multiplicity counted.The following argument principle for logharmonic mappings inD is shown in [2].Theorem 1.10 (generalized argument principle for logharmonic mappings).
LetD be a Jordan domain, and let f=hg¯ be a logharmonic mapping defined in the closure D¯ with respect to a∈B(D) satisfying |a(z)|≢1. Fix w∈ℂ such that Z(f-w,D¯)∩(∂D∪SE(D)) is empty. Then
(1.8)VZ(f-w,SL(D))-VZ(f-w,SG(D))=12π∮∂Ddarg(f-w).As a consequence of the argument principle, the following result is obtained.Theorem 1.11.
Letfn be a sequence of logharmonic mappings defined in U with respect to a given an∈B(U), where U is the unit disc. Suppose that an converges locally uniformly to a∈B(U) and that fn converges locally uniformly to a logharmonic mapping f with respect to a. If w0∉fn(U) for all n∈ℕ, then w0∉f(U).In Section2, a survey is given on univalent logharmonic mappings defined in a simply connected domain D of ℂ. Section 3 deals with univalent logharmonic mappings defined on multiply connected domains, while Section 4 considers logharmonic polynomials. The final section of the survey discusses several important subclasses of logharmonic mappings.


## 2. Univalent Logharmonic Mappings in a Simply Connected Domain

### 2.1. Motivation

LetΩ be a domain in the complex plane ℂ, and let S be a nonparametric minimal surface lying over Ω. Then S can be represented by a function s=G(u,v), w=u+iv∈Ω, and there is a univalent orientation-preserving harmonic mapping w=F(z) from an appropriate domain D of ℂ onto Ω which determines S in the following sense. The mapping F is a solution of the system of linear elliptic partial differential equation(2.1)Fz¯¯=AFz,
where A∈H(D). Since F is orientation preserving, it follows that |A(z)|<1 in D. The function A is the second dilatation of F. The value (1+|A(z)|)/(1-|A(z)|) is the quotient of the maximum value and the minimum value of the differential |dF(z)| when dz varies on the unit circle (see, e.g., [12, 13]). The representation of the minimal surface S is given by three real-valued harmonic functions (see, e.g., [13, 14]),(2.2)u(z)=Re(F(z)),v(z)=Im⁡(F(z)),s(z)=Im⁡∫zAFzdz.Since(sz)2=-Fz¯¯Fz=-A(Fz)2 in D, it follows that A belongs to H(D). In particular, each zero of A is of even order. Since the Riemannian metric of S is ds2=|Fz|2(1+|A|)2|dz|2, it follows that x=Re(z) and y=Im(z) are isothermal parameters for S. Moreover, the exterior unit normal vector n⃗(z)=(n1(z),n2(z),n3(z)),n3(z)≥0, to the minimal surface S (known as the Gauss mapping) depends only on the second dilatation function A of F. More precisely,(2.3)n⃗=(2Im⁡(A),2Re(A),1-A1+A).
The inverse of the stereographic projection of the Gauss mapping n⃗,i/A(z),is called the Weierstrass parameter.The following question arises: What are the domainsD? If φ is univalent and analytic and if F is univalent and harmonic, then the composition F∘φ (whenever well defined) is a univalent harmonic mapping but φ∘F need not be harmonic. Hence, if F represents a minimal surface over Ω (in the sense of relation (2.2)), then F(φ) represents the same minimal surface but in other isothermal parameters.Suppose thatΩ is a proper simply connected domain in ℂ. Then, we may choose for D any proper simply connected domain in ℂ. In particular, D=U or D=Ω are appropriate choices.Consider now the left half-planeD={z:Re(z)<0}, and let F be a univalent harmonic and orientation-preserving map defined in D satisfying the relation(2.4)F(z+αi)=F(z)+β∀z∈D,
where α and β are real constants. Applying the transformation (2π/β)F(2α/2π), it may be assumed without loss of generality that α=β=2π, that is,(2.5)F(z+2πi)=F(z)+2πi∀z∈D.
Whenever limx→-∞Re(F(z))=c for some c∈[-∞,∞), we will write Re(F(-∞))=c. Similarly, A(-∞)=c means that limx→-∞A(z)=c.Let UHP denote the class of all univalent harmonic orientation-preserving mappings defined on the left half-planeD={z:Re(z)<0} satisfying(2.6)F(z+2πi)=F(z)+2πi∀z∈D,Re(F(-∞))=-∞.
It follows that the second dilatation function A is periodic, that is, A(z+2πi)=A(z)+2πi in D, and therefore the Gauss map is also periodic. Observe that A(-∞) exists. Furthermore, it was shown in [6] that mappings in the class UHP admit the representation(2.7)F(z)=z+βx+H(z)+G(z)¯,
where(a)
H and G are in H(D) such that
(i)
G(-∞)=0 and H(-∞) exists and finite in ℂ,(ii)
H(z+2πi)=H(z) and G(z+2πi)=G(z) for all z∈D;(b)
(2.8)|G′(z)+β¯1+β+H′(z)|<1on⁡D,β=A(-∞)¯(1-A(-∞))1-|A(-∞)|2,andhenceRe(β)>-1.Define(2.9)f(z)=eF(log⁡(z)),z∈U.
Then f is a univalent logharmonic mapping in U with respect to a(z)=A(log(z)) and hence a∈B(U). Observe that the family of all univalent logharmonic and orientation-preserving mappings f defined in U satisfying f(0)=0 is isomorphic to the class UHP. It was shown in [4, 7] that it is easier to work with logharmonic mappings even if the differential equation becomes nonlinear.


### 2.2. Univalent Logharmonic Mappings

LetD be a simply connected domain in ℂ, D≠ℂ, and suppose that f is a univalent logharmonic mapping defined in D. If 0∉f(D), then log(f(z)) is a univalent and harmonic mapping in D. This mapping has been extensively studied in [15–18]. If f(0)=0 and f is a univalent logharmonic mapping defined in D, then the representation (1.2) of f becomes(2.10)f(z)=z|z|2βh(z)g(z)¯,
for every z∈U, where(a)
β=a(0)¯(1+a(0))/(1-|a(0)|2), and so Re(β)>-1/2,(b)
h and g are in H(U) satisfying g(0)=1 and 0∉h·g(U).It follows thatf is locally quasiconformal. The analogue of Caratheodory’s Kernel Theorem might fail for univalent logharmonic mappings. Indeed, each function(2.11)fr(z)=z(1-z)2exp⁡(-2r(Re∫0z(1+z)(1+rz)(1-z)dz)),0<r<1,
which is univalent and logharmonic with respect to ar(z)=-rz, satisfies the normalization fr(0)=0, (fr)z(0)=1, and maps the unit disc U onto the slit domain ℂ∖(-∞,-pr). The tip pr of the omitted slit varies monotonically from -1/4 to −1 as r varies from 0 to 1. The limit function limr→1fr(z)=f1(z)=(z(1-z¯))/(1-z) is univalent and logharmonic and maps U onto U. It has the boundary value f(eit)=-1 for 0<|t|≤π, and the cluster set of f1 at the point 1 is the unit circle.LetD be a simply connected domain in ℂ and z0∈D. The following characterization theorem was proved in [1].Theorem 2.1.
Letf be a univalent mapping defined in D such that f(z0)=0. Then f is of the form hg¯ if and only if f is a logharmonic mapping with respect to a∈B(D) satisfying a(z0)=m/(1+m), m∈ℕ∪{0}.Univalent logharmonic mappings have the following properties.Theorem 2.2 (see [1]).
LetD be a simply connected domain in ℂ and f a univalent logharmonic mapping defined in D with respect to a∈B(D).(a)
Thenfz(z)≠0 for all z∈D whenever f(z)≠0.(b)
Iff(z0)=0, then limz→z0(z-z0)fz(z)/f(z) exists and is in ℂ∖{0}. Therefore, (z-z0)fz(z)/f(z) is a nonvanishing function in H(D).(c)
Letα be a complex number such that Re(α)>-1/2. Then F=f|f|2α is a univalent logharmonic mapping with respect to
(2.12)a*=1+α¯1+αa+(α¯/(1+α¯))1+a(α/(1+α))∈B(D).There are few logharmonic mappings that are univalent on the whole complex planeℂ. Indeed, Abdulhadi and Bshouty [1] showed the following.Theorem 2.3.
A functionf is a univalent logharmonic mapping defined in ℂ with respect to a∈U if and only if
(2.13)f(z)=const⋅(z-z0)|z-z0|2β,β=a¯(1+a)(1-|a|2),z0∈C.Now letD be a simply connected proper domain in ℂ and f a univalent logharmonic function in D with respect to a∈B(D). Denote by φ a conformal mapping from the unit disc U onto D. Then f∘φ is univalent logharmonic in U with respect to a*=a∘φ∈B(U). Therefore, we may assume that D=U and f(0)=0.Analogous to the analytic case, we denote(2.14)SLh={f(z)=z|z|2βhg¯:fisaunivalentlogharmonicmappingdefinedinUwithh(0)=g(0)=1}.
Now 12β=1, and SLh is not compact with respect to the topology of normal convergence. Indeed, the sequence fn(z)=z|z|(1-n)/n is in SLh, and it converges uniformly to f(z)=z|z|-1 not in SLh. Our next result deals with the subclass SLh0 of SLh defined by SLh0={f∈SLh:a(0)=0 (resp., β=0)}. The following result was proved in [1].Theorem 2.4.
SLh0 is compact in the topology of normal (locally uniform) convergence.Remark 2.5.
In contrast to univalent harmonic mappings,SLh is not a normal family. Indeed,
(2.15)fn(z)=z(1-z)2|z(1-z)2|2n
is not locally uniformly bounded for n sufficiently large.The following interesting distortion theorem is due to Abdulhadi and Bshouty [1], and it was used in the proof of the mapping theorem.Theorem 2.6.
Iff∈SLh0, then |f(z)|≥|z|/4(1+|z|)2. In particular, the disc {w:|w|<1/16} is in f(U).


### 2.3. Mapping Theorem

We look for an analogue of the Riemann Mapping Theorem. LetΩ≠ℂ be a simply connected domain in ℂ, and let a∈B(U) be given. Fix z0∈U and w0∈Ω. We are interested in the existence of a univalent logharmonic function f from U into Ω with respect to the given function a and normalized by f(z0)=w0 and fz(z0)>0. If |a|≤k<1 for all z∈U, then the univalent logharmonic mappings are quasiconformal, and therefore the problem is solvable.Suppose that we want to find a univalent logharmonic mappingf with a(z)=-z, normalized by f(0)=0 and fz(0)>0 such that f maps U onto Ω=ℂ∖(-∞,-1]. Assume that such a function exists. Then, using Theorem 5.1  (α=0), it follows that f must be of the form(2.16)f=const⋅z(1-z¯)(1-z).
Observe that f is univalent in U, but maps U onto a disc, and not onto a slit domain. In other words, there is no univalent logharmonic mapping defined in U with respect to a(z)=-z satisfying f(0)=0,fz(0)>0, and f(U)=Ω. However, the following mapping theorem was proved in [1].Theorem 2.7.
LetΩ be a bounded simply connected domain in ℂ containing the origin, and whose boundary is locally connected. Let a∈B(U) be given. Then there is a univalent logharmonic function defined in U with the following properties.(i)
f is a solution of (1.1).(ii)
f(U)⊂Ω, normalized at the origin by f(z)=cz|z|2β(1+o(1)), where β=a(0)¯(1+a(0))/(1-|a(0)|2) and c>0.(iii)
limz→eitf(z)=f̂(eit) exists and is in ∂Ω for each t∈∂U∖E, where E is a countable set.(iv)
For eacheit0∈∂U,f*(eit0)=esslimt↑t0f̂(eit) and f*(eit0)=esslimt↓t0f̂(eit) exist and are in ∂Ω.(v)
Foreit0∈E, the cluster set of f at eit0 lies on a helix joining the point f*(eit0) to the point f*(eit0).Remark 2.8.
In the case where∥a∥=supz∈U|a(z)|<1, properties (ii) and (iii) imply that f(U)=Ω.Remark 2.9.
Ifeit0∈E and f*(eit0)=f*(eit0), then the cluster set at eit0 is a circle. Suppose that A=f*(eit0)≠f*(eit0)=B, then there are infinitely many helices joining A and B. But the cluster set of f at eit0 lies on one of them. For example, the cluster set of
(2.17)f(z)=z(1-z¯)(1-z)exp⁡(-2arg1-iz1-z)
at z=1 lies on the helix, γ(τ)=exp[-τ+i(π/2+τ)] joining the points f*(1)=-e-π/2 and f*(1)=-e3π/2, where the cluster set of f at z=-i is the straight line segment from f*(-i)=-e-π/2 and f*(-i)=-e3π/2.The uniqueness of the mapping theorem was proved in [6] for the special case Ω is a strictly starlike and bounded domain; that is, every ray starting at the origin intersects ∂Ω at exactly one point.Theorem 2.10 (uniqueness in the mapping theorem).
Leta∈B(U) be given such that ∥a∥=supz∈U|a(z)|<1. Let Ω be a strictly starlike and bounded domain. Then there exists a unique univalent logharmonic function f(z)=z|z|2βh(z)g(z)¯ with respect to a such that f(U)=Ω and h(0)>0.


### 2.4. Logharmonic Automorphisms

We consider univalent logharmonic mappings fromU onto U. With no loss of generality, it is assumed that f(0)=0 and h(0)>0. Otherwise, we consider an appropriate Möbius transformation of the preimage. Let AUTLh(U) denote the class of such mappings. The following two theorems established in [8] characterize completely mappings in AUTLh(U).Theorem 2.11.
Leth and g be two nonvanishing analytic functions in U. Then f(z)=z|z|2βh(z)g(z)¯ is in AUTLh(U) satisfying h(0)>0 and g(0)=1 if and only if g=1/h,Re(β)>-1/2, and Re(zh′(z))/h(z)>-1/2 in U.We now associate to eachf(z)=z|z|2βh(z)/h(z)¯ in AUTLh(U)with the mapping φ(z)=z(h(z))2∈S*.Theorem 2.12.
(a) For eachφ∈S* and for each β,  Re(β)>-1/2, there is one and only one f∈AUTLh(U) such that f(z)/(φ(z)|z|2β)>0 for every z∈U and h(0)=1.
(b) For eacha∈B(U), there is a unique solution of (1.1) which is in AUTLh(U).Remark 2.13.
Part (a) of Theorem2.12 is quite surprising. Indeed, consider φ(z)=z/(1-z)2 and β=0. Then arg(f(eit))=arg(φ(eit))=±π, almost everywhere; however, f(U)=U. To be more precise, the corresponding mapping is f(z)=z(1-z¯)/(1-z) satisfying f(eit)=-1 for all 0<|t|≤π, where the cluster set of f at the point 1 is the unit circle.


### 2.5. Univalent Logharmonic Mappings Extensions onto the Unit Disc

In 1926 Kneser [19] obtained the following result.Theorem 2.14.
LetΩ be a bounded simply connected Jordan domain, and let f* be an orientation-preserving homeomorphism from the unit disc circle ∂U onto ∂Ω. Then, if f(U)=Ω, the solution of the Dirichlet problem (the Poisson integral) is univalent on the unit disc U.Sincef(U) always contains Ω and lies in the convex hull of Ω, Kneser used Theorem 2.14 to obtain the following solution to a problem posed by Rado in [20].Theorem 2.15.
Letf* be a homeomorphism from ∂U onto ∂Ω, where Ω is a bounded convex domain. Then the Dirichlet solution f is univalent on U.In 1945, Choquet [21] independently gave another proof of Theorem 2.15, and he pointed out that it holds whenever Ω is not a convex domain.We will use the following definition.Definition 2.16.
LetD be the unit disc U or the annulus A(r,1), r∈(0,1), and suppose that f* is a continuous function defined on ∂D. One says that f is a logharmonic solution of the Dirichlet problem if(a)
f is a solution of the form (1.1),(b)
f is continuous in D,(c)
f|∂D≡f*.The next two theorems proved in [6] deal with the solutions of the Dirichlet problem for logharmonic mappings of the form (2.10).Theorem 2.17.
Letf* be a nonvanishing continuous complex-valued function defined on ∂U. Then there exist h and g analytic in U which are independent of β, such that
(2.18)f(z)=z|z|2βh(z)g(z)¯,Re(β)>-12
is a logharmonic solution of the Dirichlet problem (i.e.,f(eit)≡f*(eit)). Furthermore, if g(0)=1, then h and g are uniquely determined.Theorem 2.18.
Letf* be an orientation-preserving homeomorphism from ∂U onto ∂U,thatis,f(eit)=eiλ(t), where λ is continuous and strictly monotonically increasing on [0,2π). Furthermore, suppose that λ(2π)=λ(0)+2π. Then, for a given β with Re(β)>-1/2, the logharmonic solution of the Dirichlet problem which is of the form f(z)=z|z|2βh(z)/h(z)¯ is univalent in U.


### 2.6. Boundary Behavior

Letf be a univalent logharmonic mapping in the unit disc U with respect to a∈B(U). If |a(z)|≤k<1 for all z∈U, then f is a quasiconformal map, and its boundary behavior is the same as for conformal mappings. However, if |a| approaches one as z tends to the boundary, then the boundary behavior of f is quite different. It may happen that the boundary values are constant on an interval of ∂U, or that there are jumps as the following example shows.Example 2.19.
The functionf(z)=z(1-z¯)/(1-z) is a univalent logharmonic mapping in the unit disc U with respect to a(z)=-z, such that f(U)=U. It follows that f(eit)=-1 for all 0<|t|≤π and that the cluster set of f at the point 1 is the unit circle.The following theorem was stated in [1].Theorem 2.20.
LetΩ be a simply connected domain of ℂ whose boundary ∂Ω is locally connected, and a∈B(U). Let f be a univalent logharmonic mapping from U onto Ω satisfying f(0)=0. Then the nonrestricted limit f*(eit) of f at eit exists on ∂U∖E, where E is a countable set. If eit∈E, then f* jumps at eit, and the cluster set at eit is a subinterval of a logarithmic spiral.The next theorem [22] shows that the boundary values of f depend strongly on the values of a(eit).Theorem 2.21.
LetΩ be a simply connected domain of ℂ whose boundary ∂Ω is locally connected and a∈B(U). Suppose that the function a has an analytic extension across an open subinterval I={eit:σ<t<σ+2π} of the unit circle ∂U, such that |a(z)|≡1 in I. Let f be a univalent logharmonic mapping with respect to a which maps U onto Ω and satisfies f(0)=0. Then the following relations hold in I.(a)
Letσ<t<σ+2π and arg(f(z)) be a continuous function on the set Y:=|z:1/2<|z|<1,σ<arg(z)<τ}. If σ<t<t+h<τ, then
(2.19)log⁡(f*(ei(t+h)))-a(ei(t+h))log⁡(f*(ei(t+h)))¯-log⁡(f*(eit))+a(eit)log⁡(f*(eit))¯+∫tt+hlog⁡(f*(eiϕ))da(eiϕ)¯≡0.(b)
Iff* is continuous at eit, then
(2.20)lim⁡t↓0Im⁡a(eit)f*(ei(t+h))/f*(ei(t-h))-1h=0.(c)
Iff* jumps at eit, which must and can happen only when f*(I) lies on a segment of a logarithmic spiral, for q∈f*(I), then
(2.21)arg(log⁡f*(ei(t+0))q)=-12arg(a(eit))mod⁡π.(d)
Iff* is not constant on a subinterval of I, then the right limit
(2.22)lim⁡t↓0arg(f*(ei(t+h))f*(ei(t-h))-1)=-12arg(a(eit))mod⁡π
exists everywhere on I.


### 2.7. A Constructive Method

In this section, a method is introduced for constructing univalent logharmonic mappings from the unit disc onto a strictly starlike domainΩ, which has been successfully applied to conformal mappings (see, e.g., [23–25]), as well as for univalent harmonic mappings (see, e.g., [26, 27]).LetΩ be a strictly starlike domain of ℂ. Then ∂Ω can be expressed in the parametric form(2.23)w(t)=R(t)eit,0≤t≤2π,
where R is a positive continuous function on [0,2π]. The following notations will be used:(2.24)‖f‖∞=sup⁡{|f(z)|;z∈U},‖Ω‖∞=sup⁡{|w|;w∈Ω},d(∂Ω)=distancefromtheoriginto∂Ω.For allw∈ℂ, define(2.25)λΩ(w)={|w|R(t),0≠w=|w|eit,0,w=0.
Then(2.26)λΩ(w)<1⟺w∈Ω,λΩ(w)=1⟺w∈∂Ω,λΩ(w)>1⟺w∈C∖Ω¯,λΩ(w)=0⟺w=0.For any complex-valued functionf in U, define(2.27)μΩ(f)=sup⁡{λΩ(w):w∈f(U)}.
The following properties are due to Bshouty et al. [26].Lemma 2.22.
(a)μΩ(f)≤1⇔f(U)⊂Ω,
(b)μΩ(tf)=tμΩ(f)forallt≥0,
(c)μΩ(f)≤∥f∥∞/d(∂Ω),
(d)∥f∥∞≤μΩ(f)∥Ω∥∞,
(e)μΩ(f1+f2)≤(μΩ(f1)+μΩ(f2))(∥f∥∞/d(∂Ω)).The next lemma shows thatμΩ is lower semicontinuous with respect to the point-wise convergence; this was proved in [28].Lemma 2.23.
LetΩ be a strictly starlike domain of ℂ, and let fn be a sequence of mappings from U into ℂ which converges pointwise to f. Then limn→∞inf(μΩ(fn))≥μΩ(f). Strict inequality can hold even in the case of locally uniform convergence.LetΩ be a fixed strictly starlike domain of ℂ, and let a∈H(U),a(0)=0,|a|≤k<1 be a given (second) dilatation function. Denote by N the set of all logharmonic mappings f(z)=zh(z)g(z)¯ with respect to the given dilatation function which are normalized by g(0)=h(0)=1. Observe that β=0 since it is assumed that a(0)=0. Hengartner and Nadeau [27] solved the following optimization problem.Theorem 2.24.
LetΩ be a strictly starlike domain of ℂ, and let a∈H(U),a(0)=0, |a|≤k<1 be given. Denote by F(z)=zH(z)G(z)¯ the univalent logharmonic mapping satisfying F(U)=Ω,G(0)=1, and H(0)>0. Then there exists a unique f*∈N such that μΩ(f*)≤μΩ(f) for all f∈N and f*=F/H(0).Theorem2.24 allows us to solve the following mathematical program:(2.28)min⁡M,λ(f(z))≤M∀z∈U,∀f∈N.
For f∈N, f(z)=zh(z)g(z)¯, where h(z)=exp(∑k=1∞akzk) and(2.29)g(z)=exp⁡(∫0za(s)sds+∑k=1∞kak∫0za(s)sk-1ds).
Furthermore, each f∈N is an open mapping. Denote by Vn the set of all mappings f∈N of the form(2.30)f(z)=zexp⁡(∫0za(s)sds¯+∑k=1∞[akzk+kak¯∫0za(s)sk-1ds¯])
and by fn* any solution of the optimization problem(2.31)min⁡{μΩ(f);f∈Vn}.As a consequence of Theorem2.24, Hengartner and Rostand [28] obtained the following result.Theorem 2.25.
Leta be a polynomial such that ∥a∥∞≤k<1 in U. Then the sequence fn* of solutions of
(2.32)min⁡μΩ(f),f∈Vn,
converges locally uniformly to the univalent solution f* of
(2.33)min⁡μΩ(f),f∈N.The question remains how big couldn be. It follows from Theorem 5.24 that |an|≤2+n-1 and |bn|≤2-n-1. Suppose that Ω is a Jordan domain whose boundary ∂Ω is rectifiable and piecewise smooth. Hengartner and Nadeau [27] obtained the following additional estimate for the coefficients.Theorem 2.26.
Let(2.34)F(z)=z|z|2βexp⁡(∑k=1∞akzk+∑k=1∞bkzk¯)
be a univalent logharmonic mapping from U onto Ω, and let L be the length of F(|z|=r),0<r<1. Then
(2.35)|an|≤lim⁡r→1⁡inf⁡L(r)2πd(∂Ω)n,|bn|≤lim⁡r→1⁡inf⁡L(r)2πd(∂Ω)n.
Equality holds for the case Ω=U and f(z)=z(1-z¯)/(1-z).


## 2.1. Motivation

LetΩ be a domain in the complex plane ℂ, and let S be a nonparametric minimal surface lying over Ω. Then S can be represented by a function s=G(u,v), w=u+iv∈Ω, and there is a univalent orientation-preserving harmonic mapping w=F(z) from an appropriate domain D of ℂ onto Ω which determines S in the following sense. The mapping F is a solution of the system of linear elliptic partial differential equation(2.1)Fz¯¯=AFz,
where A∈H(D). Since F is orientation preserving, it follows that |A(z)|<1 in D. The function A is the second dilatation of F. The value (1+|A(z)|)/(1-|A(z)|) is the quotient of the maximum value and the minimum value of the differential |dF(z)| when dz varies on the unit circle (see, e.g., [12, 13]). The representation of the minimal surface S is given by three real-valued harmonic functions (see, e.g., [13, 14]),(2.2)u(z)=Re(F(z)),v(z)=Im⁡(F(z)),s(z)=Im⁡∫zAFzdz.Since(sz)2=-Fz¯¯Fz=-A(Fz)2 in D, it follows that A belongs to H(D). In particular, each zero of A is of even order. Since the Riemannian metric of S is ds2=|Fz|2(1+|A|)2|dz|2, it follows that x=Re(z) and y=Im(z) are isothermal parameters for S. Moreover, the exterior unit normal vector n⃗(z)=(n1(z),n2(z),n3(z)),n3(z)≥0, to the minimal surface S (known as the Gauss mapping) depends only on the second dilatation function A of F. More precisely,(2.3)n⃗=(2Im⁡(A),2Re(A),1-A1+A).
The inverse of the stereographic projection of the Gauss mapping n⃗,i/A(z),is called the Weierstrass parameter.The following question arises: What are the domainsD? If φ is univalent and analytic and if F is univalent and harmonic, then the composition F∘φ (whenever well defined) is a univalent harmonic mapping but φ∘F need not be harmonic. Hence, if F represents a minimal surface over Ω (in the sense of relation (2.2)), then F(φ) represents the same minimal surface but in other isothermal parameters.Suppose thatΩ is a proper simply connected domain in ℂ. Then, we may choose for D any proper simply connected domain in ℂ. In particular, D=U or D=Ω are appropriate choices.Consider now the left half-planeD={z:Re(z)<0}, and let F be a univalent harmonic and orientation-preserving map defined in D satisfying the relation(2.4)F(z+αi)=F(z)+β∀z∈D,
where α and β are real constants. Applying the transformation (2π/β)F(2α/2π), it may be assumed without loss of generality that α=β=2π, that is,(2.5)F(z+2πi)=F(z)+2πi∀z∈D.
Whenever limx→-∞Re(F(z))=c for some c∈[-∞,∞), we will write Re(F(-∞))=c. Similarly, A(-∞)=c means that limx→-∞A(z)=c.Let UHP denote the class of all univalent harmonic orientation-preserving mappings defined on the left half-planeD={z:Re(z)<0} satisfying(2.6)F(z+2πi)=F(z)+2πi∀z∈D,Re(F(-∞))=-∞.
It follows that the second dilatation function A is periodic, that is, A(z+2πi)=A(z)+2πi in D, and therefore the Gauss map is also periodic. Observe that A(-∞) exists. Furthermore, it was shown in [6] that mappings in the class UHP admit the representation(2.7)F(z)=z+βx+H(z)+G(z)¯,
where(a)
H and G are in H(D) such that
(i)
G(-∞)=0 and H(-∞) exists and finite in ℂ,(ii)
H(z+2πi)=H(z) and G(z+2πi)=G(z) for all z∈D;(b)
(2.8)|G′(z)+β¯1+β+H′(z)|<1on⁡D,β=A(-∞)¯(1-A(-∞))1-|A(-∞)|2,andhenceRe(β)>-1.Define(2.9)f(z)=eF(log⁡(z)),z∈U.
Then f is a univalent logharmonic mapping in U with respect to a(z)=A(log(z)) and hence a∈B(U). Observe that the family of all univalent logharmonic and orientation-preserving mappings f defined in U satisfying f(0)=0 is isomorphic to the class UHP. It was shown in [4, 7] that it is easier to work with logharmonic mappings even if the differential equation becomes nonlinear.


## 2.2. Univalent Logharmonic Mappings

LetD be a simply connected domain in ℂ, D≠ℂ, and suppose that f is a univalent logharmonic mapping defined in D. If 0∉f(D), then log(f(z)) is a univalent and harmonic mapping in D. This mapping has been extensively studied in [15–18]. If f(0)=0 and f is a univalent logharmonic mapping defined in D, then the representation (1.2) of f becomes(2.10)f(z)=z|z|2βh(z)g(z)¯,
for every z∈U, where(a)
β=a(0)¯(1+a(0))/(1-|a(0)|2), and so Re(β)>-1/2,(b)
h and g are in H(U) satisfying g(0)=1 and 0∉h·g(U).It follows thatf is locally quasiconformal. The analogue of Caratheodory’s Kernel Theorem might fail for univalent logharmonic mappings. Indeed, each function(2.11)fr(z)=z(1-z)2exp⁡(-2r(Re∫0z(1+z)(1+rz)(1-z)dz)),0<r<1,
which is univalent and logharmonic with respect to ar(z)=-rz, satisfies the normalization fr(0)=0, (fr)z(0)=1, and maps the unit disc U onto the slit domain ℂ∖(-∞,-pr). The tip pr of the omitted slit varies monotonically from -1/4 to −1 as r varies from 0 to 1. The limit function limr→1fr(z)=f1(z)=(z(1-z¯))/(1-z) is univalent and logharmonic and maps U onto U. It has the boundary value f(eit)=-1 for 0<|t|≤π, and the cluster set of f1 at the point 1 is the unit circle.LetD be a simply connected domain in ℂ and z0∈D. The following characterization theorem was proved in [1].Theorem 2.1.
Letf be a univalent mapping defined in D such that f(z0)=0. Then f is of the form hg¯ if and only if f is a logharmonic mapping with respect to a∈B(D) satisfying a(z0)=m/(1+m), m∈ℕ∪{0}.Univalent logharmonic mappings have the following properties.Theorem 2.2 (see [1]).
LetD be a simply connected domain in ℂ and f a univalent logharmonic mapping defined in D with respect to a∈B(D).(a)
Thenfz(z)≠0 for all z∈D whenever f(z)≠0.(b)
Iff(z0)=0, then limz→z0(z-z0)fz(z)/f(z) exists and is in ℂ∖{0}. Therefore, (z-z0)fz(z)/f(z) is a nonvanishing function in H(D).(c)
Letα be a complex number such that Re(α)>-1/2. Then F=f|f|2α is a univalent logharmonic mapping with respect to
(2.12)a*=1+α¯1+αa+(α¯/(1+α¯))1+a(α/(1+α))∈B(D).There are few logharmonic mappings that are univalent on the whole complex planeℂ. Indeed, Abdulhadi and Bshouty [1] showed the following.Theorem 2.3.
A functionf is a univalent logharmonic mapping defined in ℂ with respect to a∈U if and only if
(2.13)f(z)=const⋅(z-z0)|z-z0|2β,β=a¯(1+a)(1-|a|2),z0∈C.Now letD be a simply connected proper domain in ℂ and f a univalent logharmonic function in D with respect to a∈B(D). Denote by φ a conformal mapping from the unit disc U onto D. Then f∘φ is univalent logharmonic in U with respect to a*=a∘φ∈B(U). Therefore, we may assume that D=U and f(0)=0.Analogous to the analytic case, we denote(2.14)SLh={f(z)=z|z|2βhg¯:fisaunivalentlogharmonicmappingdefinedinUwithh(0)=g(0)=1}.
Now 12β=1, and SLh is not compact with respect to the topology of normal convergence. Indeed, the sequence fn(z)=z|z|(1-n)/n is in SLh, and it converges uniformly to f(z)=z|z|-1 not in SLh. Our next result deals with the subclass SLh0 of SLh defined by SLh0={f∈SLh:a(0)=0 (resp., β=0)}. The following result was proved in [1].Theorem 2.4.
SLh0 is compact in the topology of normal (locally uniform) convergence.Remark 2.5.
In contrast to univalent harmonic mappings,SLh is not a normal family. Indeed,
(2.15)fn(z)=z(1-z)2|z(1-z)2|2n
is not locally uniformly bounded for n sufficiently large.The following interesting distortion theorem is due to Abdulhadi and Bshouty [1], and it was used in the proof of the mapping theorem.Theorem 2.6.
Iff∈SLh0, then |f(z)|≥|z|/4(1+|z|)2. In particular, the disc {w:|w|<1/16} is in f(U).


## 2.3. Mapping Theorem

We look for an analogue of the Riemann Mapping Theorem. LetΩ≠ℂ be a simply connected domain in ℂ, and let a∈B(U) be given. Fix z0∈U and w0∈Ω. We are interested in the existence of a univalent logharmonic function f from U into Ω with respect to the given function a and normalized by f(z0)=w0 and fz(z0)>0. If |a|≤k<1 for all z∈U, then the univalent logharmonic mappings are quasiconformal, and therefore the problem is solvable.Suppose that we want to find a univalent logharmonic mappingf with a(z)=-z, normalized by f(0)=0 and fz(0)>0 such that f maps U onto Ω=ℂ∖(-∞,-1]. Assume that such a function exists. Then, using Theorem 5.1  (α=0), it follows that f must be of the form(2.16)f=const⋅z(1-z¯)(1-z).
Observe that f is univalent in U, but maps U onto a disc, and not onto a slit domain. In other words, there is no univalent logharmonic mapping defined in U with respect to a(z)=-z satisfying f(0)=0,fz(0)>0, and f(U)=Ω. However, the following mapping theorem was proved in [1].Theorem 2.7.
LetΩ be a bounded simply connected domain in ℂ containing the origin, and whose boundary is locally connected. Let a∈B(U) be given. Then there is a univalent logharmonic function defined in U with the following properties.(i)
f is a solution of (1.1).(ii)
f(U)⊂Ω, normalized at the origin by f(z)=cz|z|2β(1+o(1)), where β=a(0)¯(1+a(0))/(1-|a(0)|2) and c>0.(iii)
limz→eitf(z)=f̂(eit) exists and is in ∂Ω for each t∈∂U∖E, where E is a countable set.(iv)
For eacheit0∈∂U,f*(eit0)=esslimt↑t0f̂(eit) and f*(eit0)=esslimt↓t0f̂(eit) exist and are in ∂Ω.(v)
Foreit0∈E, the cluster set of f at eit0 lies on a helix joining the point f*(eit0) to the point f*(eit0).Remark 2.8.
In the case where∥a∥=supz∈U|a(z)|<1, properties (ii) and (iii) imply that f(U)=Ω.Remark 2.9.
Ifeit0∈E and f*(eit0)=f*(eit0), then the cluster set at eit0 is a circle. Suppose that A=f*(eit0)≠f*(eit0)=B, then there are infinitely many helices joining A and B. But the cluster set of f at eit0 lies on one of them. For example, the cluster set of
(2.17)f(z)=z(1-z¯)(1-z)exp⁡(-2arg1-iz1-z)
at z=1 lies on the helix, γ(τ)=exp[-τ+i(π/2+τ)] joining the points f*(1)=-e-π/2 and f*(1)=-e3π/2, where the cluster set of f at z=-i is the straight line segment from f*(-i)=-e-π/2 and f*(-i)=-e3π/2.The uniqueness of the mapping theorem was proved in [6] for the special case Ω is a strictly starlike and bounded domain; that is, every ray starting at the origin intersects ∂Ω at exactly one point.Theorem 2.10 (uniqueness in the mapping theorem).
Leta∈B(U) be given such that ∥a∥=supz∈U|a(z)|<1. Let Ω be a strictly starlike and bounded domain. Then there exists a unique univalent logharmonic function f(z)=z|z|2βh(z)g(z)¯ with respect to a such that f(U)=Ω and h(0)>0.


## 2.4. Logharmonic Automorphisms

We consider univalent logharmonic mappings fromU onto U. With no loss of generality, it is assumed that f(0)=0 and h(0)>0. Otherwise, we consider an appropriate Möbius transformation of the preimage. Let AUTLh(U) denote the class of such mappings. The following two theorems established in [8] characterize completely mappings in AUTLh(U).Theorem 2.11.
Leth and g be two nonvanishing analytic functions in U. Then f(z)=z|z|2βh(z)g(z)¯ is in AUTLh(U) satisfying h(0)>0 and g(0)=1 if and only if g=1/h,Re(β)>-1/2, and Re(zh′(z))/h(z)>-1/2 in U.We now associate to eachf(z)=z|z|2βh(z)/h(z)¯ in AUTLh(U)with the mapping φ(z)=z(h(z))2∈S*.Theorem 2.12.
(a) For eachφ∈S* and for each β,  Re(β)>-1/2, there is one and only one f∈AUTLh(U) such that f(z)/(φ(z)|z|2β)>0 for every z∈U and h(0)=1.
(b) For eacha∈B(U), there is a unique solution of (1.1) which is in AUTLh(U).Remark 2.13.
Part (a) of Theorem2.12 is quite surprising. Indeed, consider φ(z)=z/(1-z)2 and β=0. Then arg(f(eit))=arg(φ(eit))=±π, almost everywhere; however, f(U)=U. To be more precise, the corresponding mapping is f(z)=z(1-z¯)/(1-z) satisfying f(eit)=-1 for all 0<|t|≤π, where the cluster set of f at the point 1 is the unit circle.


## 2.5. Univalent Logharmonic Mappings Extensions onto the Unit Disc

In 1926 Kneser [19] obtained the following result.Theorem 2.14.
LetΩ be a bounded simply connected Jordan domain, and let f* be an orientation-preserving homeomorphism from the unit disc circle ∂U onto ∂Ω. Then, if f(U)=Ω, the solution of the Dirichlet problem (the Poisson integral) is univalent on the unit disc U.Sincef(U) always contains Ω and lies in the convex hull of Ω, Kneser used Theorem 2.14 to obtain the following solution to a problem posed by Rado in [20].Theorem 2.15.
Letf* be a homeomorphism from ∂U onto ∂Ω, where Ω is a bounded convex domain. Then the Dirichlet solution f is univalent on U.In 1945, Choquet [21] independently gave another proof of Theorem 2.15, and he pointed out that it holds whenever Ω is not a convex domain.We will use the following definition.Definition 2.16.
LetD be the unit disc U or the annulus A(r,1), r∈(0,1), and suppose that f* is a continuous function defined on ∂D. One says that f is a logharmonic solution of the Dirichlet problem if(a)
f is a solution of the form (1.1),(b)
f is continuous in D,(c)
f|∂D≡f*.The next two theorems proved in [6] deal with the solutions of the Dirichlet problem for logharmonic mappings of the form (2.10).Theorem 2.17.
Letf* be a nonvanishing continuous complex-valued function defined on ∂U. Then there exist h and g analytic in U which are independent of β, such that
(2.18)f(z)=z|z|2βh(z)g(z)¯,Re(β)>-12
is a logharmonic solution of the Dirichlet problem (i.e.,f(eit)≡f*(eit)). Furthermore, if g(0)=1, then h and g are uniquely determined.Theorem 2.18.
Letf* be an orientation-preserving homeomorphism from ∂U onto ∂U,thatis,f(eit)=eiλ(t), where λ is continuous and strictly monotonically increasing on [0,2π). Furthermore, suppose that λ(2π)=λ(0)+2π. Then, for a given β with Re(β)>-1/2, the logharmonic solution of the Dirichlet problem which is of the form f(z)=z|z|2βh(z)/h(z)¯ is univalent in U.


## 2.6. Boundary Behavior

Letf be a univalent logharmonic mapping in the unit disc U with respect to a∈B(U). If |a(z)|≤k<1 for all z∈U, then f is a quasiconformal map, and its boundary behavior is the same as for conformal mappings. However, if |a| approaches one as z tends to the boundary, then the boundary behavior of f is quite different. It may happen that the boundary values are constant on an interval of ∂U, or that there are jumps as the following example shows.Example 2.19.
The functionf(z)=z(1-z¯)/(1-z) is a univalent logharmonic mapping in the unit disc U with respect to a(z)=-z, such that f(U)=U. It follows that f(eit)=-1 for all 0<|t|≤π and that the cluster set of f at the point 1 is the unit circle.The following theorem was stated in [1].Theorem 2.20.
LetΩ be a simply connected domain of ℂ whose boundary ∂Ω is locally connected, and a∈B(U). Let f be a univalent logharmonic mapping from U onto Ω satisfying f(0)=0. Then the nonrestricted limit f*(eit) of f at eit exists on ∂U∖E, where E is a countable set. If eit∈E, then f* jumps at eit, and the cluster set at eit is a subinterval of a logarithmic spiral.The next theorem [22] shows that the boundary values of f depend strongly on the values of a(eit).Theorem 2.21.
LetΩ be a simply connected domain of ℂ whose boundary ∂Ω is locally connected and a∈B(U). Suppose that the function a has an analytic extension across an open subinterval I={eit:σ<t<σ+2π} of the unit circle ∂U, such that |a(z)|≡1 in I. Let f be a univalent logharmonic mapping with respect to a which maps U onto Ω and satisfies f(0)=0. Then the following relations hold in I.(a)
Letσ<t<σ+2π and arg(f(z)) be a continuous function on the set Y:=|z:1/2<|z|<1,σ<arg(z)<τ}. If σ<t<t+h<τ, then
(2.19)log⁡(f*(ei(t+h)))-a(ei(t+h))log⁡(f*(ei(t+h)))¯-log⁡(f*(eit))+a(eit)log⁡(f*(eit))¯+∫tt+hlog⁡(f*(eiϕ))da(eiϕ)¯≡0.(b)
Iff* is continuous at eit, then
(2.20)lim⁡t↓0Im⁡a(eit)f*(ei(t+h))/f*(ei(t-h))-1h=0.(c)
Iff* jumps at eit, which must and can happen only when f*(I) lies on a segment of a logarithmic spiral, for q∈f*(I), then
(2.21)arg(log⁡f*(ei(t+0))q)=-12arg(a(eit))mod⁡π.(d)
Iff* is not constant on a subinterval of I, then the right limit
(2.22)lim⁡t↓0arg(f*(ei(t+h))f*(ei(t-h))-1)=-12arg(a(eit))mod⁡π
exists everywhere on I.


## 2.7. A Constructive Method

In this section, a method is introduced for constructing univalent logharmonic mappings from the unit disc onto a strictly starlike domainΩ, which has been successfully applied to conformal mappings (see, e.g., [23–25]), as well as for univalent harmonic mappings (see, e.g., [26, 27]).LetΩ be a strictly starlike domain of ℂ. Then ∂Ω can be expressed in the parametric form(2.23)w(t)=R(t)eit,0≤t≤2π,
where R is a positive continuous function on [0,2π]. The following notations will be used:(2.24)‖f‖∞=sup⁡{|f(z)|;z∈U},‖Ω‖∞=sup⁡{|w|;w∈Ω},d(∂Ω)=distancefromtheoriginto∂Ω.For allw∈ℂ, define(2.25)λΩ(w)={|w|R(t),0≠w=|w|eit,0,w=0.
Then(2.26)λΩ(w)<1⟺w∈Ω,λΩ(w)=1⟺w∈∂Ω,λΩ(w)>1⟺w∈C∖Ω¯,λΩ(w)=0⟺w=0.For any complex-valued functionf in U, define(2.27)μΩ(f)=sup⁡{λΩ(w):w∈f(U)}.
The following properties are due to Bshouty et al. [26].Lemma 2.22.
(a)μΩ(f)≤1⇔f(U)⊂Ω,
(b)μΩ(tf)=tμΩ(f)forallt≥0,
(c)μΩ(f)≤∥f∥∞/d(∂Ω),
(d)∥f∥∞≤μΩ(f)∥Ω∥∞,
(e)μΩ(f1+f2)≤(μΩ(f1)+μΩ(f2))(∥f∥∞/d(∂Ω)).The next lemma shows thatμΩ is lower semicontinuous with respect to the point-wise convergence; this was proved in [28].Lemma 2.23.
LetΩ be a strictly starlike domain of ℂ, and let fn be a sequence of mappings from U into ℂ which converges pointwise to f. Then limn→∞inf(μΩ(fn))≥μΩ(f). Strict inequality can hold even in the case of locally uniform convergence.LetΩ be a fixed strictly starlike domain of ℂ, and let a∈H(U),a(0)=0,|a|≤k<1 be a given (second) dilatation function. Denote by N the set of all logharmonic mappings f(z)=zh(z)g(z)¯ with respect to the given dilatation function which are normalized by g(0)=h(0)=1. Observe that β=0 since it is assumed that a(0)=0. Hengartner and Nadeau [27] solved the following optimization problem.Theorem 2.24.
LetΩ be a strictly starlike domain of ℂ, and let a∈H(U),a(0)=0, |a|≤k<1 be given. Denote by F(z)=zH(z)G(z)¯ the univalent logharmonic mapping satisfying F(U)=Ω,G(0)=1, and H(0)>0. Then there exists a unique f*∈N such that μΩ(f*)≤μΩ(f) for all f∈N and f*=F/H(0).Theorem2.24 allows us to solve the following mathematical program:(2.28)min⁡M,λ(f(z))≤M∀z∈U,∀f∈N.
For f∈N, f(z)=zh(z)g(z)¯, where h(z)=exp(∑k=1∞akzk) and(2.29)g(z)=exp⁡(∫0za(s)sds+∑k=1∞kak∫0za(s)sk-1ds).
Furthermore, each f∈N is an open mapping. Denote by Vn the set of all mappings f∈N of the form(2.30)f(z)=zexp⁡(∫0za(s)sds¯+∑k=1∞[akzk+kak¯∫0za(s)sk-1ds¯])
and by fn* any solution of the optimization problem(2.31)min⁡{μΩ(f);f∈Vn}.As a consequence of Theorem2.24, Hengartner and Rostand [28] obtained the following result.Theorem 2.25.
Leta be a polynomial such that ∥a∥∞≤k<1 in U. Then the sequence fn* of solutions of
(2.32)min⁡μΩ(f),f∈Vn,
converges locally uniformly to the univalent solution f* of
(2.33)min⁡μΩ(f),f∈N.The question remains how big couldn be. It follows from Theorem 5.24 that |an|≤2+n-1 and |bn|≤2-n-1. Suppose that Ω is a Jordan domain whose boundary ∂Ω is rectifiable and piecewise smooth. Hengartner and Nadeau [27] obtained the following additional estimate for the coefficients.Theorem 2.26.
Let(2.34)F(z)=z|z|2βexp⁡(∑k=1∞akzk+∑k=1∞bkzk¯)
be a univalent logharmonic mapping from U onto Ω, and let L be the length of F(|z|=r),0<r<1. Then
(2.35)|an|≤lim⁡r→1⁡inf⁡L(r)2πd(∂Ω)n,|bn|≤lim⁡r→1⁡inf⁡L(r)2πd(∂Ω)n.
Equality holds for the case Ω=U and f(z)=z(1-z¯)/(1-z).


## 3. Univalent Logharmonic Mappings on Multiply Connected Domains

### 3.1. Univalent Logharmonic Exterior Mappings

This section considers univalent logharmonic and orientation-preserving mappingsf defined on the exterior of the unit disc U, Δ={z:|z|>1}, satisfying f(∞)=∞. These mappings are called univalent logharmonic exterior mappings. If f does not vanish on Δ, then Ψ(z)=1/f(1/z) is a univalent logharmonic mapping defined in U normalized by Ψ(0)=0. Moreover, F(ζ)=logf(eζ) is a univalent harmonic mapping defined on the right half-plane {ζ:Re(ζ)>0} satisfying the relation F(ζ+2πi)=F(ζ)+2πi and F is a solution of the linear elliptic partial differential equation(3.1)Fζ¯¯=AFζ,
where the second dilatation function A(ζ)=a(eζ),a∈B(Δ), satisfies A(ζ+2πi)=A(ζ) on {ζ:Re(ζ)>0}. Such mappings were studied in [9, 29–32]. Several authors have also studied harmonic mappings between Riemannian manifolds, and an excellent survey has been given in [33–37].The next result proved in [4] is a global representation of univalent logharmonic exterior mappings.Theorem 3.1.
Letf be a univalent logharmonic mapping defined on the exterior Δ of the closed unit disc U¯ such that f(∞)=∞.Suppose that f(p)=0 for some p∈Δ, or if f does not vanish, let p=1. Then there are two complex numbers β and γ, Re(β)>-1/2,Re(γ)>-1/2, and two nonvanishing analytic functions h and g on Δ∪{∞} such that g(∞)=1, and f is of the form
(3.2)f(z)=z|z|2β(z-p1-p¯z)|z-p1-p¯z|2γh(z)g(z)¯
for all z∈Δ.Remark 3.2.
Observe that not each function of the form (3.2) is univalent. Indeed, the function
(3.3)f(z)=z¯|z|2z-41-4z¯
is not a univalent logharmonic mapping on Δ, but it can be written in the form (3.2) by putting β=1,γ=0,p=4,h(z)=1/g(z)=(4z-1)/(4z).Letf be a univalent logharmonic exterior mapping defined on the exterior Δ of the closed unit disc U¯ such that f(∞)=∞. Applying an appropriate rotation to the preimage, we may assume that p≥1.Definition 3.3.
The class∑Lh consists of all univalent logharmonic mappings defined on Δ which are of the form (3.2), where p≥1,Re(β)>-1/2,Re(γ)>-1/2, and h and g are analytic nonvanishing functions on Δ∪{∞}, normalized by g(∞)=1 and |h(∞)|=1.Letf be a univalent logharmonic mapping in Δ with f(∞)=∞. Then there is a real number α and a positive constant A such that Af(e-αz) belongs to ∑Lh. If f does not vanish on Δ, then the set of omitted values is a continuum. In other words, there is no univalent logharmonic mapping f defined on Δ satisfying f(∞)=∞ and f(Δ)=ℂ∖{0}. Note that 0 is an exceptional point, since, for each w0∈ℂ∖{0}, there are univalent logharmonic mappings f such that f(Δ)=ℂ∖{w0}. Assume that p>1, let f∈∑Lh, and let w0 be an omitted value of f. Applying a rotation to the image f(Δ), we may assume that w0=1, and we restrict ourselves to the subclass ∑Lh*={f∈∑Lh,p>1,w0=1∉f(Δ)}.In the next theorem, Abdulhadi and Hengartner [4] gave a complete characterization of all mappings in the class ∑Lh*.Theorem 3.4.
A mappingf belongs to ∑Lh* and f(Δ)=ℂ∖{1} if and only if f is of the form
(3.4)f(z)=z¯|z|2β(z-p1-pz¯)|z-p1-pz|2γ,p>1,
where β and γ satisfy the inequality
(3.5)|β(1+γ¯)-γ(1+β¯)1+γ+γ¯-1p2-1|≤pp2-1.


### 3.2. Univalent Logharmonic Ring Mappings

In this section we investigate the familyAr of all univalent logharmonic mappings f which map an annulus A(r,1)={z:r<|z|<1},0<r<1, onto an annulus A(R,1) for some R∈[0,1) satisfying the condition(3.6)12π∫|z|=ρdargf(ρeit)=1
for all ρ∈(r,1). The last condition says that the outer boundary corresponds to the outer boundary. We call an element f∈Ar a univalent logharmonic ring mapping.Ifa≡0, then R=r and f(z)=eiαz,α∈R, are the only mappings in Ar. In the case of univalent harmonic mappings from A(r,1) onto A(R,1), it may be possible that R=0; for example, f(z)=(1-r2)-1(z-(r2/z¯)) has this property. However, Nitsche [38] has shown that there is an R0(r)<1 such that there is no univalent harmonic mapping from A(r,1) onto A(R,1) whenever R0<R<1.There is no univalent logharmonic mappings fromA(r,1), 0<r<1, onto A(0,1). This is a direct consequence of Theorem 3.5. But, on the other hand, for R there is neither a positive lower bound nor a uniform upper bound strictly less than one. Indeed, f(z)=z|z|2β,Re(β)>-1/2, is univalent on A(r,1), and its image is A(r1+2Re(β),1).Unlike the case of univalent harmonic mappings, univalent logharmonic mappings need not have a continuous extension onto the closure ofA(r,1). Indeed, f(z)=z(z¯-1)/(z-1) is a univalent logharmonic ring mapping from A(1/2,1) onto itself whose cluster sets on the outer boundary are C(f,eit)={-1}, if z=eit,0<t<2π, and C(f,1)={w:|w|=1}.LetS*(r,1) be the set of all univalent analytic functions φ on A(r,1) with the properties(i)
p(z)=zφ′(z)/φ(z)∈H(A(r,1)),(ii)
Re(p(z))>0 on A(r,1).Theorems3.5 and 3.6 [5] give a complete characterization of univalent logharmonic mappings in Ar.Theorem 3.5.
A functionf belongs to Ar if and only if
(3.7)f(z)=z|z|2βh(z)h(z)¯,
where(a)
h∈H(A(r,1)) and 0∉h(A(r,1)),(b)
Re(zh′(z)/h(z))>-1/2 on A(r,1),(c)
(1/2π)∫|z|=ρdargf(ρeit)=0,
r<ρ<1,(d)
Re(β)>-1/2.In particular, functions belonging toAr map concentric circles onto concentric circles.Theorem 3.6.
A functionf is in Ar if and only if it is of the form
(3.8)f(z)=(φ(z)|φ(z)||z|2γ),
where Re(φ)>0 and φ∈S*(r,1).Next we fix the second dilatation functiona∈H(A(r,1)),|a(z)|<1 for all z∈A(r,1). The following existence and uniqueness theorem was established in [5].Theorem 3.7.
For a givena∈H(A(r,1)),|a(z)|<1 for all z∈A(r,1), and, for a given z0∈A(r,1), there exists one and only one univalent solution f of (1.1) in Ar such that f(z0)>0.Remark 3.8.
Theorem3.7 is not true for univalent harmonic ring mappings (see [32, Theorem 7.3].)


### 3.3. Univalent Logharmonic Mappings Extensions onto the Annulus

The next two theorems proved in [6] deal with the solution of the Dirichlet problem for ring domains.Theorem 3.9.
Letf* be a nonvanishing continuous function defined on the boundary ∂A(r,1) of the annulus A(r,1). Then there exists, for each β,Re(β)>-1/2, a unique mapping f of the form (2.10), which is continuous on the closure of A(r,1) and satisfies f=f* on ∂A(r,1).Theorem 3.10.
Letf*(eit)=eiλ(t) and f*(reit)=Reiμ(t),0<R<1, be a given continuous function on ∂A(r,1),0<r<1, satisfying(a)
dλ(t)≥0 and dμ(t)≥0 on [0,2π],(b)
∫02πdλ(t)=∫02πdμ(t)=2π.
Then the logharmonic solution of the Dirichlet problem with respect tof*and A(r,1) is a univalent mapping from A(r,1) onto A(R,1).


## 3.1. Univalent Logharmonic Exterior Mappings

This section considers univalent logharmonic and orientation-preserving mappingsf defined on the exterior of the unit disc U, Δ={z:|z|>1}, satisfying f(∞)=∞. These mappings are called univalent logharmonic exterior mappings. If f does not vanish on Δ, then Ψ(z)=1/f(1/z) is a univalent logharmonic mapping defined in U normalized by Ψ(0)=0. Moreover, F(ζ)=logf(eζ) is a univalent harmonic mapping defined on the right half-plane {ζ:Re(ζ)>0} satisfying the relation F(ζ+2πi)=F(ζ)+2πi and F is a solution of the linear elliptic partial differential equation(3.1)Fζ¯¯=AFζ,
where the second dilatation function A(ζ)=a(eζ),a∈B(Δ), satisfies A(ζ+2πi)=A(ζ) on {ζ:Re(ζ)>0}. Such mappings were studied in [9, 29–32]. Several authors have also studied harmonic mappings between Riemannian manifolds, and an excellent survey has been given in [33–37].The next result proved in [4] is a global representation of univalent logharmonic exterior mappings.Theorem 3.1.
Letf be a univalent logharmonic mapping defined on the exterior Δ of the closed unit disc U¯ such that f(∞)=∞.Suppose that f(p)=0 for some p∈Δ, or if f does not vanish, let p=1. Then there are two complex numbers β and γ, Re(β)>-1/2,Re(γ)>-1/2, and two nonvanishing analytic functions h and g on Δ∪{∞} such that g(∞)=1, and f is of the form
(3.2)f(z)=z|z|2β(z-p1-p¯z)|z-p1-p¯z|2γh(z)g(z)¯
for all z∈Δ.Remark 3.2.
Observe that not each function of the form (3.2) is univalent. Indeed, the function
(3.3)f(z)=z¯|z|2z-41-4z¯
is not a univalent logharmonic mapping on Δ, but it can be written in the form (3.2) by putting β=1,γ=0,p=4,h(z)=1/g(z)=(4z-1)/(4z).Letf be a univalent logharmonic exterior mapping defined on the exterior Δ of the closed unit disc U¯ such that f(∞)=∞. Applying an appropriate rotation to the preimage, we may assume that p≥1.Definition 3.3.
The class∑Lh consists of all univalent logharmonic mappings defined on Δ which are of the form (3.2), where p≥1,Re(β)>-1/2,Re(γ)>-1/2, and h and g are analytic nonvanishing functions on Δ∪{∞}, normalized by g(∞)=1 and |h(∞)|=1.Letf be a univalent logharmonic mapping in Δ with f(∞)=∞. Then there is a real number α and a positive constant A such that Af(e-αz) belongs to ∑Lh. If f does not vanish on Δ, then the set of omitted values is a continuum. In other words, there is no univalent logharmonic mapping f defined on Δ satisfying f(∞)=∞ and f(Δ)=ℂ∖{0}. Note that 0 is an exceptional point, since, for each w0∈ℂ∖{0}, there are univalent logharmonic mappings f such that f(Δ)=ℂ∖{w0}. Assume that p>1, let f∈∑Lh, and let w0 be an omitted value of f. Applying a rotation to the image f(Δ), we may assume that w0=1, and we restrict ourselves to the subclass ∑Lh*={f∈∑Lh,p>1,w0=1∉f(Δ)}.In the next theorem, Abdulhadi and Hengartner [4] gave a complete characterization of all mappings in the class ∑Lh*.Theorem 3.4.
A mappingf belongs to ∑Lh* and f(Δ)=ℂ∖{1} if and only if f is of the form
(3.4)f(z)=z¯|z|2β(z-p1-pz¯)|z-p1-pz|2γ,p>1,
where β and γ satisfy the inequality
(3.5)|β(1+γ¯)-γ(1+β¯)1+γ+γ¯-1p2-1|≤pp2-1.


## 3.2. Univalent Logharmonic Ring Mappings

In this section we investigate the familyAr of all univalent logharmonic mappings f which map an annulus A(r,1)={z:r<|z|<1},0<r<1, onto an annulus A(R,1) for some R∈[0,1) satisfying the condition(3.6)12π∫|z|=ρdargf(ρeit)=1
for all ρ∈(r,1). The last condition says that the outer boundary corresponds to the outer boundary. We call an element f∈Ar a univalent logharmonic ring mapping.Ifa≡0, then R=r and f(z)=eiαz,α∈R, are the only mappings in Ar. In the case of univalent harmonic mappings from A(r,1) onto A(R,1), it may be possible that R=0; for example, f(z)=(1-r2)-1(z-(r2/z¯)) has this property. However, Nitsche [38] has shown that there is an R0(r)<1 such that there is no univalent harmonic mapping from A(r,1) onto A(R,1) whenever R0<R<1.There is no univalent logharmonic mappings fromA(r,1), 0<r<1, onto A(0,1). This is a direct consequence of Theorem 3.5. But, on the other hand, for R there is neither a positive lower bound nor a uniform upper bound strictly less than one. Indeed, f(z)=z|z|2β,Re(β)>-1/2, is univalent on A(r,1), and its image is A(r1+2Re(β),1).Unlike the case of univalent harmonic mappings, univalent logharmonic mappings need not have a continuous extension onto the closure ofA(r,1). Indeed, f(z)=z(z¯-1)/(z-1) is a univalent logharmonic ring mapping from A(1/2,1) onto itself whose cluster sets on the outer boundary are C(f,eit)={-1}, if z=eit,0<t<2π, and C(f,1)={w:|w|=1}.LetS*(r,1) be the set of all univalent analytic functions φ on A(r,1) with the properties(i)
p(z)=zφ′(z)/φ(z)∈H(A(r,1)),(ii)
Re(p(z))>0 on A(r,1).Theorems3.5 and 3.6 [5] give a complete characterization of univalent logharmonic mappings in Ar.Theorem 3.5.
A functionf belongs to Ar if and only if
(3.7)f(z)=z|z|2βh(z)h(z)¯,
where(a)
h∈H(A(r,1)) and 0∉h(A(r,1)),(b)
Re(zh′(z)/h(z))>-1/2 on A(r,1),(c)
(1/2π)∫|z|=ρdargf(ρeit)=0,
r<ρ<1,(d)
Re(β)>-1/2.In particular, functions belonging toAr map concentric circles onto concentric circles.Theorem 3.6.
A functionf is in Ar if and only if it is of the form
(3.8)f(z)=(φ(z)|φ(z)||z|2γ),
where Re(φ)>0 and φ∈S*(r,1).Next we fix the second dilatation functiona∈H(A(r,1)),|a(z)|<1 for all z∈A(r,1). The following existence and uniqueness theorem was established in [5].Theorem 3.7.
For a givena∈H(A(r,1)),|a(z)|<1 for all z∈A(r,1), and, for a given z0∈A(r,1), there exists one and only one univalent solution f of (1.1) in Ar such that f(z0)>0.Remark 3.8.
Theorem3.7 is not true for univalent harmonic ring mappings (see [32, Theorem 7.3].)


## 3.3. Univalent Logharmonic Mappings Extensions onto the Annulus

The next two theorems proved in [6] deal with the solution of the Dirichlet problem for ring domains.Theorem 3.9.
Letf* be a nonvanishing continuous function defined on the boundary ∂A(r,1) of the annulus A(r,1). Then there exists, for each β,Re(β)>-1/2, a unique mapping f of the form (2.10), which is continuous on the closure of A(r,1) and satisfies f=f* on ∂A(r,1).Theorem 3.10.
Letf*(eit)=eiλ(t) and f*(reit)=Reiμ(t),0<R<1, be a given continuous function on ∂A(r,1),0<r<1, satisfying(a)
dλ(t)≥0 and dμ(t)≥0 on [0,2π],(b)
∫02πdλ(t)=∫02πdμ(t)=2π.
Then the logharmonic solution of the Dirichlet problem with respect tof*and A(r,1) is a univalent mapping from A(r,1) onto A(R,1).


## 4. Logharmonic Polynomials

Denote bypn an analytic polynomial of degree n. A logharmonic polynomial is a function of the form f=pnpm¯. In contrast to the analytic case, there are nonconstant logharmonic polynomials which are not p-valent for every p>0. For example, the function f(z)=zz¯ is a logharmonic polynomial in ℂ with respect to a=-1. Moreover, the function f(z)=(z-1)(z¯+1) is a logharmonic polynomial in ℂ with respect to a(z)=(z+1)/(z-1). This polynomial is two-valent and omits the half-plane Re(w)<-1. On the other hand, they inherit the property limz→∞f(z)=∞ of analytic polynomials. This follows from the fact that |f|=|pnpm¯|=|pnpm|. However, the converse is not true; there are logharmonic functions f=hg¯ defined in ℂ which are not logharmonic polynomials and have the property limz→∞f(z)=∞. The function f(z)=zeze-z¯ is such an example. Note that there are harmonic polynomials pn(z)+pm(z)¯ which do not satisfy limz→∞f(z)=∞. However, if it is assumed that a(∞) exists and |a(∞)|≠1, then the following result [2] is deduced.Theorem 4.1.
Letf=hg¯ be a logharmonic function in ℂ such that limz→∞f(z)=∞. If limz→∞a(z)=a(∞) exists and if |a(∞)|≠1, then f is a polynomial.Denote byNZ(f-w,D) the cardinality of Z(f-w,D), that is, the number of zeros of f-w in D, multiplicity not counted. The polynomial f(z)=|z|2 has the property that NZ(f-1,ℂ)=∞. On the other hand, using Theorem 2.3, it follows that a univalent logharmonic mapping in ℂ is necessarily a polynomial which is either of the form f(z)=const·(z-a)(z-a)m¯ or of its conjugate, where const≠0,a∈ℂ, and m=0,1,2,…. There are functions of the form f=hg¯ which are not polynomials but have the property that NZ(f-w,ℂ) is finite and uniformly bounded for all w∈ℂ. For example, the function f(z)=zezez¯-w has at most two zeros for all fixed w∈ℂ. The following result was shown in [2].Theorem 4.2.
Letf=hg¯ be a logharmonic function in ℂ such that NZ(f-w,G) is finite for at least two different values of w, limz→∞a(z)=a(∞) exists with |a(∞)|≠1, then f is a polynomial.An upper bound on the number ofw-points of a logharmonic polynomial can be readily obtained by using Bezout’s theorem [39].Theorem 4.3 (see [39]).
Letp(x,y) and q(x,y) be polynomials in the real variables x and y with real coefficients. If deg(p)=n and deg(q)=m, then either p and q have at most nm common zeros or they have infinitely many zeros.Wilmshurst [40] has shown that Bezout’s theorem gives a sharp upper bound for the number of zeros of a harmonic polynomial and hence for polyanalytic polynomials (see, e.g.,  [41, 42]). However, this is not true for logharmonic polynomials.Letf=pnpm¯ be a logharmonic polynomial of degree n+m. Then f(z)-w=∑k=0n∑j=0makjzkz¯j. The functions P(z)=Re(f(z)) and Q(z)=Im(f(z)) are real-valued polynomials in x and y and are of degree n+m. Applying Bezout’s theorem, we conclude with the following estimate.Theorem 4.4.
Letf=pnpm¯ be a logharmonic polynomial defined in ℂ. Then either f-w has infinitely many zeros or f-w has at most (n+m)2 zeros for all w∈ℂ.The bound is not the best possible. Indeed, a quadratic polynomial is of the formf(z)=p2(z),  f(z)¯=p2(z), or f(z)=a(z+b)(z+c)¯. In all three cases, f-w has either infinitely many zeros or it has at most two.Observe that the logharmonic polynomialf(z)=(z-1)/(z¯+1) is 2-valent and omits the half-plane Re(w)<-1 and that |a|≢1. However, the situation changes if |a(∞)|≠1 and we have the following result [2].Theorem 4.5.
Letf=pnpm¯ be a logharmonic polynomial defined in ℂ, and suppose that n>m. Fix w∈ℂ such that Z(f-w,ℂ)∩(∂D∪SE(D)) is empty. Then the number of zeros VZ(f-w,SE(ℂ)) of f-w, and hence also the valency V(f,ℂ) of f in ℂ, is at least n-m. The bound is best possible.The following result is an immediate consequence of Theorem4.5.Corollary 4.6.
Letf=pnpm¯ be a logharmonic polynomial defined in ℂ, and suppose that n>m. Then(i)
f(ℂ)=ℂ,(ii)
for almost allw∈ℂ, the function f-w has at least n-m disjoint zeros.The next result characterizes polynomials of finite valency [2].Theorem 4.7.
Letf=pnpm¯ be a logharmonic polynomial defined in ℂ, such that pn≢const·pm. Then the cardinality NZ(f-w,ℂ) of the zero set Z(f-w,ℂ) is finite (hence, by Bezout’s theorem, uniformly bounded) for all w∈ℂ.Remark 4.8.
Ifpn≡const·pm, then the image lies on a straight line.


## 5. Subclasses of Logharmonic Mappings

### 5.1. Spirallike Logharmonic Mappings

LetΩ be a simply connected domain if ℂ contains the origin. We say that Ω is α-spirallike,  -π/2<α<π/2, if w∈Ω implies that wexp(-teiα)∈Ω for all t≥0. If α=0, the domain Ω is called starlike (with respect to the origin). We will use the following notations.(a)
SLhα is the set of all univalent logharmonic mappings f in U satisfying f(0)=0, h(0)=g(0)=1, and f(U) is an α-spirallike domain.(b)
Sα={f∈SLhα and f∈H(U)}.(c)
SLh*=SLh0 and S*=S0, for which f(U) is starlike (with respect to the origin).To eachf(z)=z|z|2βh(z)g(z)¯∈SLhα, we associate the analytic function ψ(z)=zh(z)/g(z)eiα, ψ(0)=0. Abdulhadi and Hengartner [8] gave a representation theorem for mappings in the class SLhα.Theorem 5.1.
(a) Iff∈SLhα, then ψ∈Sα.
(b)
For any givenψ∈Sα and a∈B(U), there are h and g in H(U) uniquely determined such that
(i)
0∉h·g(U),  h(0)=g(0)=1,(ii)
ψ(z)=zh(z)/g(z)eiα,(iii)
f(z)=z|z|2βh(z)g(z)¯ is a solution of (1.1) in SLhα, where β=(a(0)¯(1+a(0)))/(1-|a(0)|2).Remark 5.2.
Theorem5.1 has no equivalence for the class of all convex univalent logharmonic mappings. Indeed, ψ(z)=z is a convex mapping, a(z)=z4∈B(U), but f(z)=z/|1-z4|1/2 is not a convex mapping.Remark 5.3.
Theorem5.1 asserts that the class SLhα,α fixed in (-π/2,π/2), is isomorphic to Sα×B(U).The following result is an immediate consequence of Theorem5.1.Corollary 5.4.
Iff∈SLhα, then f(rz)/r∈SLhα  for all r∈(0,1). In other words, level sets inherit the property of being α-spirallike.The next result is an integral representation forf∈SLhα [8].Theorem 5.5.
A functionf∈SLhα if and only if there are two probability measures μ and ν on the Borel sets of ∂U and an a(0)∈U such that
(5.1)f(z)=z|z|2β⋅exp⁡{∫∂U×∂UK(z,ζ,ξ;a(0))dμ(ζ)dν(ξ)},
where
(5.2)β=a(0)¯(1+a(0))1-|a(0)|2,K(z,ζ,ξ;a(0))=-2cos⁡(α)⋅eiα⋅log⁡(1-ζz)+2eiαRe{eiαlog⁡(1-ζz)}+T(z,ζ,ξ;a(0)),T(z,ζ,ξ;a(0))=2eiαReeiα(1+a(0))(1-a(0)¯e-2iα)ζ+e-iα(1+a(0)¯)(1-a(0)e2iα)ξ(ζ-ξ)|1-a(0)e2iα|2×log⁡1-ξz1-ζz,
if |ζ|=|ξ|=1,ζ≠ξ, and
(5.3)T(z,ζ,ζ;a(0))=4cos⁡(α)⋅eiα⋅Reζz(1-ζz)1-|a(0)|2|1-a(0)e2iα|2.Observe thatSLhα is not compact, but Theorem 5.5 can be used to solve extremal problems over the class of mappings in SLhα with a given a(0)=0.We have seen in Corollary5.4 that if f is a univalent logharmonic mapping in U, f(0)=0, and if f(U) is starlike, then f(|z|<r) is starlike (with respect to the origin) for all r∈(0,1). The next result proved in [8] shows that this property may fail whenever f(0)≠0.Theorem 5.6.
For eachz0∈U∖{0}, there are univalent logharmonic mappings f such that f(z0)=0,f(U) is starlike (with respect to the origin), but no level set f(|z|<r),|z0|=ρ<r<1, is starlike.


### 5.2. Close-to-Starlike Logharmonic Mappings

#### 5.2.1. Logharmonic Mappings with Positive Real Part

LetPLh be the set of all logharmonic mappings R in U which are of the form R=HG¯, where H and G are in H(U), H(0)=G(0)=1, such that Re(R(z))>0 for all z∈U. In particular, the set P of all analytic functions p in U with p(0)=1 and Re(p(z))>0 in U is a subset of PLh.The next result [43] describes the connection between PLh and P.Theorem 5.7.
A functionR=HG¯∈PLh if and only if p=H/G∈P.
As a consequence of Theorem5.7, it follows that R admits the representation
(5.4)R(z)=p(z)exp⁡2Re∫0za(s)1-a(s)p′(s)p(s)ds,
where a∈B(U) and p is an analytic function with positive real part normalized by p(0)=1.The following result [43] is a distortion theorem for the class PLh.Theorem 5.8.
LetR(z)=H(z)G(z)¯∈PLh, and suppose that a(0)=0. Then for z∈U(i)
exp(-2|z|/(1-|z|))≤|R(z)|≤exp(2|z|/(1-|z|)),(ii)
|Rz(z)|≤(2/(1-|z|)(1-z|2))exp(2|z|/(1-|z|)),(iii)
|Rz¯(z)|≤(2|z|/(1-|z|)(1-|z|2))exp(2|z|/(1-|z|)).
Equality occurs for the right inequalities ifR is a function of the form R0(ζz), |ζ|=1, where(5.5)R0(z)=1+z1-z|1+z1-z|exp⁡(Re2z1-z),
and equality occurs for the left inequalities if R is of the form
(5.6)1R0(ζz),|ζ|=1.


#### 5.2.2. Close-to-Starlike Logharmonic Mappings

LetF(z)=z|z|2βhg¯ be a logharmonic mapping. The function F is close to starlike if F is a product between a starlike logharmonic mapping f(z)=z|z|2βh*g*¯∈SLh* which is a solution of (1.1) with respect to a∈B(U) and a logharmonic mapping with positive real part R∈PLh with the same second dilatation function a.The geometric interpretation for a close-to-starlike logharmonic mappings is that the radius vector of the image of|z|=r<1 never turns back by an amount more than π.Denote byCSTLh the set of all close-to-starlike logharmonic mappings. It contains in particular the set CST of all analytic close-to-starlike functions which was introduced by Reade [44] in 1955. Also, the set SLh* of all starlike univalent logharmonic mappings is a subset of CSTLh (take R(z)≡1 in the product). Furthermore, if F(z)=z|z|2βhg¯ is a logharmonic mapping with respect to a∈B(U) satisfying h(0)=g(0)=1 and ReF(z)/z|z|2β>0, then F is a close-to-starlike logharmonic mapping, where(5.7)f(z)=z|z|2β|exp⁡(∫0za(s)/s1-a(s)ds)|2.
On the other hand, a mapping F∈CSTLh need not necessarily be univalent. For example, take F(z)=z(1+z), where z∈S* and 1+z∈P.Our next result is a representation theorem for the classCSTLh proved in [43].Theorem 5.9.
(a) LetF=z|z|2βhg¯ be in CSTLh. Then ψ=zh/g∈CST.
(b)
Given anyψ∈CST and a∈B(U), there are h and g in H(U) uniquely determined such that
(i)
0∉h·g(U),h(0)=g(0)=1,(ii)
ψ(z)=zh/g,(iii)
F=z|z|2βhg¯ is in CSTLh which is a solution of (1.1) with respect to the given a.Corollary 5.10.
F∈CSTLh if and only if F(rz)/r∈CSTLh for all r∈(0,1).In the next result the radius of univalence and the radius of starlikeness are determined for those mappings in the setCSTLh [43].Theorem 5.11.
LetF=z|z|2βhg¯∈CSTLh. Then F maps the disc |z|<R, R≤2-3, onto a starlike domain. The upper bound is best possible for all a∈B(U).Combining Theorems5.8 and 5.11 with α=0, we obtain the following distortion theorem for the class CSTLh.Theorem 5.12.
LetF=zhg¯∈CSTLh. Then, for every z∈U,(a)
|z|exp(-2|z|/(1-|z|)-4|z|/(1+|z|))≤|F(z)|≤|z|exp(6|z|/(1-|z|)),(b)
|Fz(z)|≤((|z|2+4|z|+1)/(1-|z|)2(1+|z|))exp(6|z|/(1-|z|)),(c)
|Fz¯(z)|≤(|z|(|z|2+4|z|+1)/(1-|z|)2(1+|z|))exp(6|z|/(1-|z|)).
Equality holds for the right inequalities ifF is a function of the form(5.8)Fη,ζ(z)=z(1-ηz¯)(1-ηz)1+ζz1-ζz|1-ζz1+ζz|exp⁡(Re[4ηz1-ηz+2ζz1-ζz]),
where |η|=|ζ|=1, and for the left inequalities if F is a function of the form
(5.9)Fη,ζ(z)=z(1-ηz¯)(1-ηz)1+ζz1-ζz|1-ζz1+ζz|exp⁡(Re[4ηz1-ηz-2ζz1-ζz]),
where |η|=|ζ|=1.


### 5.3. Typically Real Logharmonic Mappings

A logharmonic mappingf is said to be typically real if and only if f is real whenever z is real and if f is normalized by f(0)=0 and h(0)g(0)¯=1, or equivalently by f(0)=0 and h(0)=g(0)=1. Denote by TLh the class of all orientation-preserving typically real logharmonic mappings. Since f is orientation preserving and univalent on the interval (-1,1), it follows that f is of the form (2.10). Furthermore, if f∈TLh, then β (and hence, also a(0)) has to be real and yields the relation(5.10)Im⁡zIm⁡f(z)>0,∀z∈U∖R.
The class TLh is a compact convex set with respect to the topology of locally uniform convergence, and it contains, in particular, the set T of all analytic typically real functions.


#### 5.3.1. Basic Properties of Mappings fromTLh

The following representation theorem for typically real logharmonic mappings was proved in [45].Theorem 5.13.
(a) Iff(z)=z|z|2βh(z)g(z)¯ is in TLh, then ϕ=zh/g∈T.
(b)
Givenϕ∈T and a∈B(U) such that β∈ℝ and a(0)∈ℝ, there are uniquely determined mappings h and g in H(U) such that
(i)
0∉h·g(U),h(0)=g(0)=1,(ii)
ϕ(z)=zh/g,(iii)
F=z|z|2βhg¯ is in TLh which is a solution of (1.1) with respect to the given a.As a consequence of Theorem5.13, it follows that(5.11)f(z)=zh(z)g(z)¯=ϕ(z)|g(z)|2,
where(5.12)g(z)=exp⁡∫0za(s)ϕ′(s)(1-a(s))ϕ(s)ds,zh(z)=ϕ(z)g(z).Denote byTLh0 the subclass of TLh  (β=0) consisting of all mappings F from TLh for which ϕ=zh/g=z/(1-z2). Then F is of the form(5.13)F(z)=z1-z2exp⁡2Re∫0za(s)(1+s2)s(1-a(s))(1-s2)ds.The next theorem links the classTLh with the class PLh.Theorem 5.14 (see [45]).
Letf(z)=zh(z)g(z)¯∈TLh with respect to a∈B(U), and a(0)=0. Then there exist an  R∈PLh and an F∈TLh0, such that both functions are logharmonic with respect to the same a and
(5.14)f(z)=F(z)R(z).The next result is a distortion theorem for the classTLh0.Theorem 5.15 (see [45]).
LetF(z)=zh(z)g(z)¯∈TLh0. Then, for z∈U,(a)
|F(z)|≤|z|exp(2|z|/(1-|z|)),(b)
|Fz(z)|≤((1+|z|2)/(1-|z|2)(1-|z|))exp(2|z|/(1-|z|)),(c)
|Fz¯(z)|≤(|z|(1+|z|2)/(1-|z|2)(1-|z|))exp(2|z|/(1-|z|)).
Equality holds if and only ifF is of the form η¯F0(ηz),  |η|=1, where(5.15)F0(z)=z1-z2|1-z2|exp⁡(Re(2z1-z)).Combining Theorems5.8, 5.14, and 5.15, the following distortion theorem is obtained for the class TLh.Theorem 5.16.
Letf(z)=zh(z)g(z)¯∈TLh. Then, for z∈U,(a)
|f(z)|≤|z|exp(4|z|/(1-|z|)),(b)
|fz(z)|≤((1+|z|)/(1-|z|2))exp(4|z|/(1-|z|)),(c)
|fz¯(z)|≤(|z|(1+|z|)/(1-|z|2))exp(4|z|/(1-|z|)).
Equality holds iff is of the form η¯f0(ηz),|η|=1, where
(5.16)f0(z)=z(1-z̅)1-zexp⁡⁡(Re(4z1-z)).Remark 5.17.
The functionf0 given in (5.16) plays the role of the Koebe mapping in the set of logharmonic mappings (see, e.g., [1, 6]).The next result gives the radius of univalence and the radius of starlikeness for the mappings in the setTLh [45].Theorem 5.18 (see [45]).
Letf(z)=z|z|2βh(z)g(z)¯∈TLh. Then f maps the disc {z:|z|<R0}, where R0=(1+5-2+25)/2, onto a starlike domain. The upper bound is the best possible for all a∈B(U).


#### 5.3.2. Univalent Mappings inTLh

Now we consider univalent mappings inTLh. For analytic typically real functions, it is known that if t(z)=z+∑n=2∞anzn is univalent in the unit disc U, then t belongs to T if and only if the image t(U) is a domain symmetric with respect to the real axis.One might consider a similar problem inTLh. Let f(z)=z|z|2βh(z)g(z)¯ be a univalent logharmonic mapping in the unit disc U, and h(0)=g(0)=1, β>-1/2. Observe that β (and hence a(0)) is real. Is it true that f belongs to TLh if and only if the image of f(U) is a symmetric domain with respect to the real axis?The answer is negative in both directions. Indeed, the function(5.17)f(z)=z(1+iz8)(1-iz¯8)
is a normalized univalent logharmonic typically real mapping, but f(U) is not symmetric with respect to the real axis. On the other hand, the function f(z)=z(1+iz¯)/(1-iz) is a univalent logharmonic mapping from U onto U, and f(U) is symmetric with respect to the real axis, but f is not typically real (for more details, see [45]).Additional conditions ona and on the image domain Ω=f(U) are needed in order to get an affirmative answer to the question posed above.Theorem 5.19 (see [45]).
Letf(z)=z|z|2βh(z)g(z)¯  be a univalent (orientation-preserving) logharmonic mapping in the unit disc U and normalized by f(0)=0, h(0)=g(0)¯=1. Suppose that the second dilatation function a has real coefficients, that is, a(z)≡a(z¯)¯.(Observe that the condition a(0) real or equivalently β real is automatically satisfied.)(a)
Iff is typically real, then f(U) is symmetric with respect to the real axis.(b)
If|a|≤k<1 in U and f(U) is a strictly starlike Jordan domain symmetric with respect to the real axis, then f is typically real.


### 5.4. Starlike Logharmonic Mappings of Orderα

Letf=z|z|2βhg¯ be a univalent logharmonic mapping. We say that f is starlike logharmonic mapping of order α if(5.18)∂argf(reiθ)∂θ=Rezfz-z¯fz¯f>α,0≤α<1,
for all z∈U. Denote by STLh(α) the set of all starlike logharmonic mappings of order α. If α=0, we get the class of starlike logharmonic mappings. Also, let ST(α)={f∈STLh(α) and f∈H(U)}.In this section, we obtain two representation theorems [46] for functions in STLh(α). In the first, we establish the connection between the classes STLh(α) and ST(α). The second is an integral representation theorem.Theorem 5.20.
Letf(z)=zh(z)g(z)¯ be a logharmonic mapping in U, 0∉hg(U). Then f∈STLh(α) if and only if φ(z)=zh(z)/g(z)∈ST(α).Theorem 5.21.
A functionf=zhg¯∈STLh(α) with a(0)=0 if and only if there are two probability measures μ and ν such that
(5.19)f(z)=zexp⁡(∫∂U×∂UK(z,ζ,ξ)dμ(ζ)dν(ξ)),
where
(5.20)K(z,ζ,ξ)=(1-α)log⁡(1+ζz¯1-ζz)+T(z,ζ,ξ),T(z,ζ,ξ)={-2(1-α)Im⁡(ζ+ξζ-ξ)arg(1-ξz1-ζz)-2αlog⁡|1-ξz|,if|ζ|=|ξ|=1,ζ≠ξ,(1-α)Re(4ζz1-ζz)-2αlog⁡|1-ζz|,if|ζ|=|ξ|=1,ζ=ξ.Remark 5.22.
Theorem5.21 can be used to solve extremal problems for the class STLh(α) with a(0)=0. For example, see Theorem 5.23.The following is a distortion theorem for the classSTLh(α) with a(0)=0.Theorem 5.23 (see [46]).
Letf=zh(z)g(z)¯∈STLh(α) with a(0)=0. Then, for z∈U,
(5.21)|z|(1+|z|)2αexp⁡((1-α)-4|z|1+|z|)≤|f(z)|≤|z|(1-|z|)2αexp⁡((1-α)4|z|1-|z|).
Equalities occur if and only if f(z)=ζ¯f0(ζz), |ζ|=1, where
(5.22)f0(z)=z(1-z¯1-z)1(1-z¯)2αexp⁡((1-α)Re4z1-z).The next result gives sharp coefficient estimates for functionsh and g in the starlike logharmonic mapping f=zh(z)g(z)¯.Theorem 5.24 (see [6]).
Letf=zh(z)g(z)¯∈STLh(0) with a(0)=0, and put
(5.23)h(z)=exp⁡(∑k=1∞akzk),g(z)=exp⁡(∑k=1∞bkzk).
Then
(5.24)|an|≤2+1n,|bn|≤2-1n
for all n≥1. Equality holds for the mapping
(5.25)f(z)=z1-zeiα¯1-zeiαexp⁡(4zeiα1-zeiα),α∈(0,2π).


### 5.5. Functions with Logharmonic Laplacian

We consider the class of all continuous complex-valued functionsF=u+iv in a domain D⊆C such that the Laplacian of F is logharmonic. Note that log(ΔF) is harmonic in D if it satisfies theLaplace equation Δ(log(ΔF))=0, where(5.26)Δ=4∂2∂z∂z¯.
In any simply connected domain D, we can write(5.27)F=r2L+H,z=reiθ,
where L is logharmonic and H is harmonic in D. It is known that L and H can be expressed as(5.28)L=h1g1¯,H=h2+g2¯,
where h1,g1,h2, and g2 are analytic in D. Denote by LLh(U) the set of all functions of the form (5.27) which are defined in the unit disc U.Note that the compositionL∘ϕ of a logharmonic function L with an analytic function ϕ is logharmonic and, also, the composition H∘ϕ of a harmonic function H with analytic function ϕ is harmonic, while this is not true for the function F. Also, if F1(z)=r2L1(z) and F2(z)=r2L2(z) are in LLh(U), where L1 and L2 are logharmonic with respect to the same a, then F1αF2β, α+β=1, is also in LLh(U).Denote the Jacobian ofW by JW. Then(5.29)JW=|Wz|2-|Wz¯|2.
Also let(5.30)λW=|Wz|-|Wz¯|,ΛW=|Wz|+|Wz¯|.
Then JW=λWΛW.


#### 5.5.1. The Univalence of Functions with Logharmonic Laplacian

First a lower bound for the area of the range ofF(z)=r2L(z) is established, where L is a starlike univalent logharmonic mapping.Theorem 5.25 (see [47]).
LetF(z)=r2L(z), where L=hg¯ is starlike univalent logharmonic in U, with g(0)=1 and h′(0)=1. Let A(r,F) denote the area of F(Ur), where Ur={z:|z|<r}, for r<1. Then
(5.31)A(r,F)≥2π[-2r+r2-2r33+r42-r55+r66-r88+2ln⁡(1+r)].
Equality holds if and only if L0(z)=r2z(1+z¯/2)/(1+z/2) or one of its rotations.Definition 5.26.
LetL be logharmonic function in U. A complex-valued function of the form F(z)=r2L(z) is starlike in U if it is orientation preserving, F(0)=0,F(z)≠0 when z≠0 and the curve F(reit) is starlike with respect to the origin for each 0<r<1. In other words, ∂argF(reit)/∂t=Re((zFz-z¯Fz¯)/F)>0.Remark 5.27.
Note that starlike functions are univalent inU.The following theorem links starlike functions inLLh(U) with the class of starlike analytic functions.Theorem 5.28 (see [48]).
LetF(z)=r2L(z), where L(z)=h(z)g(z)¯, be a logharmonic function in U with respect to a, where a∈B(U) with a(0)=0. Then F is starlike univalent in U if and only if ψ(z)=h(z)/g(z) is starlike univalent function in U.Corollary 5.29.
The functionr2L(z) is starlike for all conformal starlike functions L.A characterization of the logharmonic Laplacian solutions of the Dirichlet problem in the unit discU is given in [48].Theorem 5.30.
LetF* be an orientation-preserving homeomorphism from ∂U onto ∂U, that is, F*(eit)=eiλ(t), where λ is continuous and strictly monotonically increasing on [0,2π]. Furthermore, suppose that λ(2π)=λ(0)+2π. Then F(z)=z¯|z|2h(z)/h(z)¯ is a univalent solution of the Dirichlet problem in U.For the general caseF(z)=r2L(z)+H(z), a sufficient condition is obtained that makes F locally univalent.Theorem 5.31 (see [48]).
LetF(z)=r2h1(z)g1(z)¯+h2(z)+g2(z)¯ be in the class LLh(U). Suppose that ψ(z)=h1(z)/g1(z) is starlike univalent in U, and |g2′(z)|<|h2′(z)| for z∈U. If
(5.32)Re[g2′(r2h1g1¯)z¯¯]<Re[h2′(r2h1g1¯)z¯],
then JF(z)>0  for z≠0, and F is locally univalent.


#### 5.5.2. Landau’s Theorem for Functions with Logharmonic Laplacian

Lewy’s famous theorem [49] states that a harmonic function W is locally univalent in D (univalent in some neighborhood of each point in D) if and only if its Jacobian does not vanish in D.The classical Landau Theorem states that iff is analytic in the unit disc U with f(0)=0,f′(0)=1, and |f(z)|<M for z∈U, then f is univalent in the disc Uρ0={z:|z|<ρ0} with(5.33)ρ0=1M+M2-1,
and f(Uρ0) contains a disc UR0 with R0=Mρ02. This result is sharp, with the extremal function f(z)=Mz(1-Mz)/(M-z) (see [19]).Chen et al. [50] obtained a version of Landau’s Theorem for bounded harmonic mappings of the unit disc. Unfortunately their result is not sharp. Better estimates were given in [51] and later in [52].Specifically, it was shown in [52] that if f is harmonic in the unit disc U with f(0)=0,Jf(0)=1, and |f(z)|<M for z∈U, then f is univalent in the disc Uρ1={z:|z|<ρ1} with(5.34)ρ1=1-22Mπ+8M2,
and f(Uρ1) contains a disc UR1 with R1=π/4M-2M(ρ12/(1-ρ1)). This result is the best known, but not sharp.The following Schwarz lemma for harmonic mappings is due to Grigoryan [52].Lemma 5.32 (Schwarz lemma).
Letf be a harmonic mapping of the unit disc U with f(0)=0 and f(U)⊂U. Then
(5.35)|f(z)|≤4πarctan|z|≤4π|z|,Λf(0)≤4π.Recently Mao et al. [53] established the Schwarz lemma for logharmonic mappings, through which two versions of Landau’s theorem for these mappings were obtained.The next theorem gives Landau’s theorem for functions with logharmonic Laplacian of the formF=r2L(z).Theorem 5.33 (see [47]).
LetL be logharmonic in U such that L(0)=0, JL(0)=1, and |L(z)|<M for z∈U. Then there is a constant 0<ρ2<1 such that F=r2L is univalent in the disc |z|<ρ2, ρ2 is the solution of the equation 1=2ρ2M/(1-ρ22)-2Mρ2/(1-ρ22)2, and f(Uρ2) contains a disc UR2 with R2=ρ22-2Mρ24/(1-ρ22). This result is not sharp.Finally we give a Landau theorem for functions of logharmonic Laplacian of the formF=r2L+K.Theorem 5.34 (see [47]).
LetF=r2L+K,z=reiθ be in LLh(U), where L is logharmonic and K is harmonic in the unit disc U, such that L(0)=K(0)=0,JF(0)=1, and |L| and |K| are both bounded by M. Then there is a constant 0<ρ3<1 such that F is univalent in |z|<ρ3. Specifically ρ3satisfies
(5.36)π4M-2ρ3M-2M(ρ33(1-ρ32)2+1(1-ρ3)2-1)=0,
and F(Uρ3) contains a disc UR3, where
(5.37)R3=π4Mρ3-ρ32M11-ρ32-2Mρ321-ρ3.


## 5.1. Spirallike Logharmonic Mappings

LetΩ be a simply connected domain if ℂ contains the origin. We say that Ω is α-spirallike,  -π/2<α<π/2, if w∈Ω implies that wexp(-teiα)∈Ω for all t≥0. If α=0, the domain Ω is called starlike (with respect to the origin). We will use the following notations.(a)
SLhα is the set of all univalent logharmonic mappings f in U satisfying f(0)=0, h(0)=g(0)=1, and f(U) is an α-spirallike domain.(b)
Sα={f∈SLhα and f∈H(U)}.(c)
SLh*=SLh0 and S*=S0, for which f(U) is starlike (with respect to the origin).To eachf(z)=z|z|2βh(z)g(z)¯∈SLhα, we associate the analytic function ψ(z)=zh(z)/g(z)eiα, ψ(0)=0. Abdulhadi and Hengartner [8] gave a representation theorem for mappings in the class SLhα.Theorem 5.1.
(a) Iff∈SLhα, then ψ∈Sα.
(b)
For any givenψ∈Sα and a∈B(U), there are h and g in H(U) uniquely determined such that
(i)
0∉h·g(U),  h(0)=g(0)=1,(ii)
ψ(z)=zh(z)/g(z)eiα,(iii)
f(z)=z|z|2βh(z)g(z)¯ is a solution of (1.1) in SLhα, where β=(a(0)¯(1+a(0)))/(1-|a(0)|2).Remark 5.2.
Theorem5.1 has no equivalence for the class of all convex univalent logharmonic mappings. Indeed, ψ(z)=z is a convex mapping, a(z)=z4∈B(U), but f(z)=z/|1-z4|1/2 is not a convex mapping.Remark 5.3.
Theorem5.1 asserts that the class SLhα,α fixed in (-π/2,π/2), is isomorphic to Sα×B(U).The following result is an immediate consequence of Theorem5.1.Corollary 5.4.
Iff∈SLhα, then f(rz)/r∈SLhα  for all r∈(0,1). In other words, level sets inherit the property of being α-spirallike.The next result is an integral representation forf∈SLhα [8].Theorem 5.5.
A functionf∈SLhα if and only if there are two probability measures μ and ν on the Borel sets of ∂U and an a(0)∈U such that
(5.1)f(z)=z|z|2β⋅exp⁡{∫∂U×∂UK(z,ζ,ξ;a(0))dμ(ζ)dν(ξ)},
where
(5.2)β=a(0)¯(1+a(0))1-|a(0)|2,K(z,ζ,ξ;a(0))=-2cos⁡(α)⋅eiα⋅log⁡(1-ζz)+2eiαRe{eiαlog⁡(1-ζz)}+T(z,ζ,ξ;a(0)),T(z,ζ,ξ;a(0))=2eiαReeiα(1+a(0))(1-a(0)¯e-2iα)ζ+e-iα(1+a(0)¯)(1-a(0)e2iα)ξ(ζ-ξ)|1-a(0)e2iα|2×log⁡1-ξz1-ζz,
if |ζ|=|ξ|=1,ζ≠ξ, and
(5.3)T(z,ζ,ζ;a(0))=4cos⁡(α)⋅eiα⋅Reζz(1-ζz)1-|a(0)|2|1-a(0)e2iα|2.Observe thatSLhα is not compact, but Theorem 5.5 can be used to solve extremal problems over the class of mappings in SLhα with a given a(0)=0.We have seen in Corollary5.4 that if f is a univalent logharmonic mapping in U, f(0)=0, and if f(U) is starlike, then f(|z|<r) is starlike (with respect to the origin) for all r∈(0,1). The next result proved in [8] shows that this property may fail whenever f(0)≠0.Theorem 5.6.
For eachz0∈U∖{0}, there are univalent logharmonic mappings f such that f(z0)=0,f(U) is starlike (with respect to the origin), but no level set f(|z|<r),|z0|=ρ<r<1, is starlike.


## 5.2. Close-to-Starlike Logharmonic Mappings

### 5.2.1. Logharmonic Mappings with Positive Real Part

LetPLh be the set of all logharmonic mappings R in U which are of the form R=HG¯, where H and G are in H(U), H(0)=G(0)=1, such that Re(R(z))>0 for all z∈U. In particular, the set P of all analytic functions p in U with p(0)=1 and Re(p(z))>0 in U is a subset of PLh.The next result [43] describes the connection between PLh and P.Theorem 5.7.
A functionR=HG¯∈PLh if and only if p=H/G∈P.
As a consequence of Theorem5.7, it follows that R admits the representation
(5.4)R(z)=p(z)exp⁡2Re∫0za(s)1-a(s)p′(s)p(s)ds,
where a∈B(U) and p is an analytic function with positive real part normalized by p(0)=1.The following result [43] is a distortion theorem for the class PLh.Theorem 5.8.
LetR(z)=H(z)G(z)¯∈PLh, and suppose that a(0)=0. Then for z∈U(i)
exp(-2|z|/(1-|z|))≤|R(z)|≤exp(2|z|/(1-|z|)),(ii)
|Rz(z)|≤(2/(1-|z|)(1-z|2))exp(2|z|/(1-|z|)),(iii)
|Rz¯(z)|≤(2|z|/(1-|z|)(1-|z|2))exp(2|z|/(1-|z|)).
Equality occurs for the right inequalities ifR is a function of the form R0(ζz), |ζ|=1, where(5.5)R0(z)=1+z1-z|1+z1-z|exp⁡(Re2z1-z),
and equality occurs for the left inequalities if R is of the form
(5.6)1R0(ζz),|ζ|=1.


### 5.2.2. Close-to-Starlike Logharmonic Mappings

LetF(z)=z|z|2βhg¯ be a logharmonic mapping. The function F is close to starlike if F is a product between a starlike logharmonic mapping f(z)=z|z|2βh*g*¯∈SLh* which is a solution of (1.1) with respect to a∈B(U) and a logharmonic mapping with positive real part R∈PLh with the same second dilatation function a.The geometric interpretation for a close-to-starlike logharmonic mappings is that the radius vector of the image of|z|=r<1 never turns back by an amount more than π.Denote byCSTLh the set of all close-to-starlike logharmonic mappings. It contains in particular the set CST of all analytic close-to-starlike functions which was introduced by Reade [44] in 1955. Also, the set SLh* of all starlike univalent logharmonic mappings is a subset of CSTLh (take R(z)≡1 in the product). Furthermore, if F(z)=z|z|2βhg¯ is a logharmonic mapping with respect to a∈B(U) satisfying h(0)=g(0)=1 and ReF(z)/z|z|2β>0, then F is a close-to-starlike logharmonic mapping, where(5.7)f(z)=z|z|2β|exp⁡(∫0za(s)/s1-a(s)ds)|2.
On the other hand, a mapping F∈CSTLh need not necessarily be univalent. For example, take F(z)=z(1+z), where z∈S* and 1+z∈P.Our next result is a representation theorem for the classCSTLh proved in [43].Theorem 5.9.
(a) LetF=z|z|2βhg¯ be in CSTLh. Then ψ=zh/g∈CST.
(b)
Given anyψ∈CST and a∈B(U), there are h and g in H(U) uniquely determined such that
(i)
0∉h·g(U),h(0)=g(0)=1,(ii)
ψ(z)=zh/g,(iii)
F=z|z|2βhg¯ is in CSTLh which is a solution of (1.1) with respect to the given a.Corollary 5.10.
F∈CSTLh if and only if F(rz)/r∈CSTLh for all r∈(0,1).In the next result the radius of univalence and the radius of starlikeness are determined for those mappings in the setCSTLh [43].Theorem 5.11.
LetF=z|z|2βhg¯∈CSTLh. Then F maps the disc |z|<R, R≤2-3, onto a starlike domain. The upper bound is best possible for all a∈B(U).Combining Theorems5.8 and 5.11 with α=0, we obtain the following distortion theorem for the class CSTLh.Theorem 5.12.
LetF=zhg¯∈CSTLh. Then, for every z∈U,(a)
|z|exp(-2|z|/(1-|z|)-4|z|/(1+|z|))≤|F(z)|≤|z|exp(6|z|/(1-|z|)),(b)
|Fz(z)|≤((|z|2+4|z|+1)/(1-|z|)2(1+|z|))exp(6|z|/(1-|z|)),(c)
|Fz¯(z)|≤(|z|(|z|2+4|z|+1)/(1-|z|)2(1+|z|))exp(6|z|/(1-|z|)).
Equality holds for the right inequalities ifF is a function of the form(5.8)Fη,ζ(z)=z(1-ηz¯)(1-ηz)1+ζz1-ζz|1-ζz1+ζz|exp⁡(Re[4ηz1-ηz+2ζz1-ζz]),
where |η|=|ζ|=1, and for the left inequalities if F is a function of the form
(5.9)Fη,ζ(z)=z(1-ηz¯)(1-ηz)1+ζz1-ζz|1-ζz1+ζz|exp⁡(Re[4ηz1-ηz-2ζz1-ζz]),
where |η|=|ζ|=1.


## 5.2.1. Logharmonic Mappings with Positive Real Part

LetPLh be the set of all logharmonic mappings R in U which are of the form R=HG¯, where H and G are in H(U), H(0)=G(0)=1, such that Re(R(z))>0 for all z∈U. In particular, the set P of all analytic functions p in U with p(0)=1 and Re(p(z))>0 in U is a subset of PLh.The next result [43] describes the connection between PLh and P.Theorem 5.7.
A functionR=HG¯∈PLh if and only if p=H/G∈P.
As a consequence of Theorem5.7, it follows that R admits the representation
(5.4)R(z)=p(z)exp⁡2Re∫0za(s)1-a(s)p′(s)p(s)ds,
where a∈B(U) and p is an analytic function with positive real part normalized by p(0)=1.The following result [43] is a distortion theorem for the class PLh.Theorem 5.8.
LetR(z)=H(z)G(z)¯∈PLh, and suppose that a(0)=0. Then for z∈U(i)
exp(-2|z|/(1-|z|))≤|R(z)|≤exp(2|z|/(1-|z|)),(ii)
|Rz(z)|≤(2/(1-|z|)(1-z|2))exp(2|z|/(1-|z|)),(iii)
|Rz¯(z)|≤(2|z|/(1-|z|)(1-|z|2))exp(2|z|/(1-|z|)).
Equality occurs for the right inequalities ifR is a function of the form R0(ζz), |ζ|=1, where(5.5)R0(z)=1+z1-z|1+z1-z|exp⁡(Re2z1-z),
and equality occurs for the left inequalities if R is of the form
(5.6)1R0(ζz),|ζ|=1.


## 5.2.2. Close-to-Starlike Logharmonic Mappings

LetF(z)=z|z|2βhg¯ be a logharmonic mapping. The function F is close to starlike if F is a product between a starlike logharmonic mapping f(z)=z|z|2βh*g*¯∈SLh* which is a solution of (1.1) with respect to a∈B(U) and a logharmonic mapping with positive real part R∈PLh with the same second dilatation function a.The geometric interpretation for a close-to-starlike logharmonic mappings is that the radius vector of the image of|z|=r<1 never turns back by an amount more than π.Denote byCSTLh the set of all close-to-starlike logharmonic mappings. It contains in particular the set CST of all analytic close-to-starlike functions which was introduced by Reade [44] in 1955. Also, the set SLh* of all starlike univalent logharmonic mappings is a subset of CSTLh (take R(z)≡1 in the product). Furthermore, if F(z)=z|z|2βhg¯ is a logharmonic mapping with respect to a∈B(U) satisfying h(0)=g(0)=1 and ReF(z)/z|z|2β>0, then F is a close-to-starlike logharmonic mapping, where(5.7)f(z)=z|z|2β|exp⁡(∫0za(s)/s1-a(s)ds)|2.
On the other hand, a mapping F∈CSTLh need not necessarily be univalent. For example, take F(z)=z(1+z), where z∈S* and 1+z∈P.Our next result is a representation theorem for the classCSTLh proved in [43].Theorem 5.9.
(a) LetF=z|z|2βhg¯ be in CSTLh. Then ψ=zh/g∈CST.
(b)
Given anyψ∈CST and a∈B(U), there are h and g in H(U) uniquely determined such that
(i)
0∉h·g(U),h(0)=g(0)=1,(ii)
ψ(z)=zh/g,(iii)
F=z|z|2βhg¯ is in CSTLh which is a solution of (1.1) with respect to the given a.Corollary 5.10.
F∈CSTLh if and only if F(rz)/r∈CSTLh for all r∈(0,1).In the next result the radius of univalence and the radius of starlikeness are determined for those mappings in the setCSTLh [43].Theorem 5.11.
LetF=z|z|2βhg¯∈CSTLh. Then F maps the disc |z|<R, R≤2-3, onto a starlike domain. The upper bound is best possible for all a∈B(U).Combining Theorems5.8 and 5.11 with α=0, we obtain the following distortion theorem for the class CSTLh.Theorem 5.12.
LetF=zhg¯∈CSTLh. Then, for every z∈U,(a)
|z|exp(-2|z|/(1-|z|)-4|z|/(1+|z|))≤|F(z)|≤|z|exp(6|z|/(1-|z|)),(b)
|Fz(z)|≤((|z|2+4|z|+1)/(1-|z|)2(1+|z|))exp(6|z|/(1-|z|)),(c)
|Fz¯(z)|≤(|z|(|z|2+4|z|+1)/(1-|z|)2(1+|z|))exp(6|z|/(1-|z|)).
Equality holds for the right inequalities ifF is a function of the form(5.8)Fη,ζ(z)=z(1-ηz¯)(1-ηz)1+ζz1-ζz|1-ζz1+ζz|exp⁡(Re[4ηz1-ηz+2ζz1-ζz]),
where |η|=|ζ|=1, and for the left inequalities if F is a function of the form
(5.9)Fη,ζ(z)=z(1-ηz¯)(1-ηz)1+ζz1-ζz|1-ζz1+ζz|exp⁡(Re[4ηz1-ηz-2ζz1-ζz]),
where |η|=|ζ|=1.


## 5.3. Typically Real Logharmonic Mappings

A logharmonic mappingf is said to be typically real if and only if f is real whenever z is real and if f is normalized by f(0)=0 and h(0)g(0)¯=1, or equivalently by f(0)=0 and h(0)=g(0)=1. Denote by TLh the class of all orientation-preserving typically real logharmonic mappings. Since f is orientation preserving and univalent on the interval (-1,1), it follows that f is of the form (2.10). Furthermore, if f∈TLh, then β (and hence, also a(0)) has to be real and yields the relation(5.10)Im⁡zIm⁡f(z)>0,∀z∈U∖R.
The class TLh is a compact convex set with respect to the topology of locally uniform convergence, and it contains, in particular, the set T of all analytic typically real functions.


### 5.3.1. Basic Properties of Mappings fromTLh

The following representation theorem for typically real logharmonic mappings was proved in [45].Theorem 5.13.
(a) Iff(z)=z|z|2βh(z)g(z)¯ is in TLh, then ϕ=zh/g∈T.
(b)
Givenϕ∈T and a∈B(U) such that β∈ℝ and a(0)∈ℝ, there are uniquely determined mappings h and g in H(U) such that
(i)
0∉h·g(U),h(0)=g(0)=1,(ii)
ϕ(z)=zh/g,(iii)
F=z|z|2βhg¯ is in TLh which is a solution of (1.1) with respect to the given a.As a consequence of Theorem5.13, it follows that(5.11)f(z)=zh(z)g(z)¯=ϕ(z)|g(z)|2,
where(5.12)g(z)=exp⁡∫0za(s)ϕ′(s)(1-a(s))ϕ(s)ds,zh(z)=ϕ(z)g(z).Denote byTLh0 the subclass of TLh  (β=0) consisting of all mappings F from TLh for which ϕ=zh/g=z/(1-z2). Then F is of the form(5.13)F(z)=z1-z2exp⁡2Re∫0za(s)(1+s2)s(1-a(s))(1-s2)ds.The next theorem links the classTLh with the class PLh.Theorem 5.14 (see [45]).
Letf(z)=zh(z)g(z)¯∈TLh with respect to a∈B(U), and a(0)=0. Then there exist an  R∈PLh and an F∈TLh0, such that both functions are logharmonic with respect to the same a and
(5.14)f(z)=F(z)R(z).The next result is a distortion theorem for the classTLh0.Theorem 5.15 (see [45]).
LetF(z)=zh(z)g(z)¯∈TLh0. Then, for z∈U,(a)
|F(z)|≤|z|exp(2|z|/(1-|z|)),(b)
|Fz(z)|≤((1+|z|2)/(1-|z|2)(1-|z|))exp(2|z|/(1-|z|)),(c)
|Fz¯(z)|≤(|z|(1+|z|2)/(1-|z|2)(1-|z|))exp(2|z|/(1-|z|)).
Equality holds if and only ifF is of the form η¯F0(ηz),  |η|=1, where(5.15)F0(z)=z1-z2|1-z2|exp⁡(Re(2z1-z)).Combining Theorems5.8, 5.14, and 5.15, the following distortion theorem is obtained for the class TLh.Theorem 5.16.
Letf(z)=zh(z)g(z)¯∈TLh. Then, for z∈U,(a)
|f(z)|≤|z|exp(4|z|/(1-|z|)),(b)
|fz(z)|≤((1+|z|)/(1-|z|2))exp(4|z|/(1-|z|)),(c)
|fz¯(z)|≤(|z|(1+|z|)/(1-|z|2))exp(4|z|/(1-|z|)).
Equality holds iff is of the form η¯f0(ηz),|η|=1, where
(5.16)f0(z)=z(1-z̅)1-zexp⁡⁡(Re(4z1-z)).Remark 5.17.
The functionf0 given in (5.16) plays the role of the Koebe mapping in the set of logharmonic mappings (see, e.g., [1, 6]).The next result gives the radius of univalence and the radius of starlikeness for the mappings in the setTLh [45].Theorem 5.18 (see [45]).
Letf(z)=z|z|2βh(z)g(z)¯∈TLh. Then f maps the disc {z:|z|<R0}, where R0=(1+5-2+25)/2, onto a starlike domain. The upper bound is the best possible for all a∈B(U).


### 5.3.2. Univalent Mappings inTLh

Now we consider univalent mappings inTLh. For analytic typically real functions, it is known that if t(z)=z+∑n=2∞anzn is univalent in the unit disc U, then t belongs to T if and only if the image t(U) is a domain symmetric with respect to the real axis.One might consider a similar problem inTLh. Let f(z)=z|z|2βh(z)g(z)¯ be a univalent logharmonic mapping in the unit disc U, and h(0)=g(0)=1, β>-1/2. Observe that β (and hence a(0)) is real. Is it true that f belongs to TLh if and only if the image of f(U) is a symmetric domain with respect to the real axis?The answer is negative in both directions. Indeed, the function(5.17)f(z)=z(1+iz8)(1-iz¯8)
is a normalized univalent logharmonic typically real mapping, but f(U) is not symmetric with respect to the real axis. On the other hand, the function f(z)=z(1+iz¯)/(1-iz) is a univalent logharmonic mapping from U onto U, and f(U) is symmetric with respect to the real axis, but f is not typically real (for more details, see [45]).Additional conditions ona and on the image domain Ω=f(U) are needed in order to get an affirmative answer to the question posed above.Theorem 5.19 (see [45]).
Letf(z)=z|z|2βh(z)g(z)¯  be a univalent (orientation-preserving) logharmonic mapping in the unit disc U and normalized by f(0)=0, h(0)=g(0)¯=1. Suppose that the second dilatation function a has real coefficients, that is, a(z)≡a(z¯)¯.(Observe that the condition a(0) real or equivalently β real is automatically satisfied.)(a)
Iff is typically real, then f(U) is symmetric with respect to the real axis.(b)
If|a|≤k<1 in U and f(U) is a strictly starlike Jordan domain symmetric with respect to the real axis, then f is typically real.


## 5.3.1. Basic Properties of Mappings fromTLh

The following representation theorem for typically real logharmonic mappings was proved in [45].Theorem 5.13.
(a) Iff(z)=z|z|2βh(z)g(z)¯ is in TLh, then ϕ=zh/g∈T.
(b)
Givenϕ∈T and a∈B(U) such that β∈ℝ and a(0)∈ℝ, there are uniquely determined mappings h and g in H(U) such that
(i)
0∉h·g(U),h(0)=g(0)=1,(ii)
ϕ(z)=zh/g,(iii)
F=z|z|2βhg¯ is in TLh which is a solution of (1.1) with respect to the given a.As a consequence of Theorem5.13, it follows that(5.11)f(z)=zh(z)g(z)¯=ϕ(z)|g(z)|2,
where(5.12)g(z)=exp⁡∫0za(s)ϕ′(s)(1-a(s))ϕ(s)ds,zh(z)=ϕ(z)g(z).Denote byTLh0 the subclass of TLh  (β=0) consisting of all mappings F from TLh for which ϕ=zh/g=z/(1-z2). Then F is of the form(5.13)F(z)=z1-z2exp⁡2Re∫0za(s)(1+s2)s(1-a(s))(1-s2)ds.The next theorem links the classTLh with the class PLh.Theorem 5.14 (see [45]).
Letf(z)=zh(z)g(z)¯∈TLh with respect to a∈B(U), and a(0)=0. Then there exist an  R∈PLh and an F∈TLh0, such that both functions are logharmonic with respect to the same a and
(5.14)f(z)=F(z)R(z).The next result is a distortion theorem for the classTLh0.Theorem 5.15 (see [45]).
LetF(z)=zh(z)g(z)¯∈TLh0. Then, for z∈U,(a)
|F(z)|≤|z|exp(2|z|/(1-|z|)),(b)
|Fz(z)|≤((1+|z|2)/(1-|z|2)(1-|z|))exp(2|z|/(1-|z|)),(c)
|Fz¯(z)|≤(|z|(1+|z|2)/(1-|z|2)(1-|z|))exp(2|z|/(1-|z|)).
Equality holds if and only ifF is of the form η¯F0(ηz),  |η|=1, where(5.15)F0(z)=z1-z2|1-z2|exp⁡(Re(2z1-z)).Combining Theorems5.8, 5.14, and 5.15, the following distortion theorem is obtained for the class TLh.Theorem 5.16.
Letf(z)=zh(z)g(z)¯∈TLh. Then, for z∈U,(a)
|f(z)|≤|z|exp(4|z|/(1-|z|)),(b)
|fz(z)|≤((1+|z|)/(1-|z|2))exp(4|z|/(1-|z|)),(c)
|fz¯(z)|≤(|z|(1+|z|)/(1-|z|2))exp(4|z|/(1-|z|)).
Equality holds iff is of the form η¯f0(ηz),|η|=1, where
(5.16)f0(z)=z(1-z̅)1-zexp⁡⁡(Re(4z1-z)).Remark 5.17.
The functionf0 given in (5.16) plays the role of the Koebe mapping in the set of logharmonic mappings (see, e.g., [1, 6]).The next result gives the radius of univalence and the radius of starlikeness for the mappings in the setTLh [45].Theorem 5.18 (see [45]).
Letf(z)=z|z|2βh(z)g(z)¯∈TLh. Then f maps the disc {z:|z|<R0}, where R0=(1+5-2+25)/2, onto a starlike domain. The upper bound is the best possible for all a∈B(U).


## 5.3.2. Univalent Mappings inTLh

Now we consider univalent mappings inTLh. For analytic typically real functions, it is known that if t(z)=z+∑n=2∞anzn is univalent in the unit disc U, then t belongs to T if and only if the image t(U) is a domain symmetric with respect to the real axis.One might consider a similar problem inTLh. Let f(z)=z|z|2βh(z)g(z)¯ be a univalent logharmonic mapping in the unit disc U, and h(0)=g(0)=1, β>-1/2. Observe that β (and hence a(0)) is real. Is it true that f belongs to TLh if and only if the image of f(U) is a symmetric domain with respect to the real axis?The answer is negative in both directions. Indeed, the function(5.17)f(z)=z(1+iz8)(1-iz¯8)
is a normalized univalent logharmonic typically real mapping, but f(U) is not symmetric with respect to the real axis. On the other hand, the function f(z)=z(1+iz¯)/(1-iz) is a univalent logharmonic mapping from U onto U, and f(U) is symmetric with respect to the real axis, but f is not typically real (for more details, see [45]).Additional conditions ona and on the image domain Ω=f(U) are needed in order to get an affirmative answer to the question posed above.Theorem 5.19 (see [45]).
Letf(z)=z|z|2βh(z)g(z)¯  be a univalent (orientation-preserving) logharmonic mapping in the unit disc U and normalized by f(0)=0, h(0)=g(0)¯=1. Suppose that the second dilatation function a has real coefficients, that is, a(z)≡a(z¯)¯.(Observe that the condition a(0) real or equivalently β real is automatically satisfied.)(a)
Iff is typically real, then f(U) is symmetric with respect to the real axis.(b)
If|a|≤k<1 in U and f(U) is a strictly starlike Jordan domain symmetric with respect to the real axis, then f is typically real.


## 5.4. Starlike Logharmonic Mappings of Orderα

Letf=z|z|2βhg¯ be a univalent logharmonic mapping. We say that f is starlike logharmonic mapping of order α if(5.18)∂argf(reiθ)∂θ=Rezfz-z¯fz¯f>α,0≤α<1,
for all z∈U. Denote by STLh(α) the set of all starlike logharmonic mappings of order α. If α=0, we get the class of starlike logharmonic mappings. Also, let ST(α)={f∈STLh(α) and f∈H(U)}.In this section, we obtain two representation theorems [46] for functions in STLh(α). In the first, we establish the connection between the classes STLh(α) and ST(α). The second is an integral representation theorem.Theorem 5.20.
Letf(z)=zh(z)g(z)¯ be a logharmonic mapping in U, 0∉hg(U). Then f∈STLh(α) if and only if φ(z)=zh(z)/g(z)∈ST(α).Theorem 5.21.
A functionf=zhg¯∈STLh(α) with a(0)=0 if and only if there are two probability measures μ and ν such that
(5.19)f(z)=zexp⁡(∫∂U×∂UK(z,ζ,ξ)dμ(ζ)dν(ξ)),
where
(5.20)K(z,ζ,ξ)=(1-α)log⁡(1+ζz¯1-ζz)+T(z,ζ,ξ),T(z,ζ,ξ)={-2(1-α)Im⁡(ζ+ξζ-ξ)arg(1-ξz1-ζz)-2αlog⁡|1-ξz|,if|ζ|=|ξ|=1,ζ≠ξ,(1-α)Re(4ζz1-ζz)-2αlog⁡|1-ζz|,if|ζ|=|ξ|=1,ζ=ξ.Remark 5.22.
Theorem5.21 can be used to solve extremal problems for the class STLh(α) with a(0)=0. For example, see Theorem 5.23.The following is a distortion theorem for the classSTLh(α) with a(0)=0.Theorem 5.23 (see [46]).
Letf=zh(z)g(z)¯∈STLh(α) with a(0)=0. Then, for z∈U,
(5.21)|z|(1+|z|)2αexp⁡((1-α)-4|z|1+|z|)≤|f(z)|≤|z|(1-|z|)2αexp⁡((1-α)4|z|1-|z|).
Equalities occur if and only if f(z)=ζ¯f0(ζz), |ζ|=1, where
(5.22)f0(z)=z(1-z¯1-z)1(1-z¯)2αexp⁡((1-α)Re4z1-z).The next result gives sharp coefficient estimates for functionsh and g in the starlike logharmonic mapping f=zh(z)g(z)¯.Theorem 5.24 (see [6]).
Letf=zh(z)g(z)¯∈STLh(0) with a(0)=0, and put
(5.23)h(z)=exp⁡(∑k=1∞akzk),g(z)=exp⁡(∑k=1∞bkzk).
Then
(5.24)|an|≤2+1n,|bn|≤2-1n
for all n≥1. Equality holds for the mapping
(5.25)f(z)=z1-zeiα¯1-zeiαexp⁡(4zeiα1-zeiα),α∈(0,2π).


## 5.5. Functions with Logharmonic Laplacian

We consider the class of all continuous complex-valued functionsF=u+iv in a domain D⊆C such that the Laplacian of F is logharmonic. Note that log(ΔF) is harmonic in D if it satisfies theLaplace equation Δ(log(ΔF))=0, where(5.26)Δ=4∂2∂z∂z¯.
In any simply connected domain D, we can write(5.27)F=r2L+H,z=reiθ,
where L is logharmonic and H is harmonic in D. It is known that L and H can be expressed as(5.28)L=h1g1¯,H=h2+g2¯,
where h1,g1,h2, and g2 are analytic in D. Denote by LLh(U) the set of all functions of the form (5.27) which are defined in the unit disc U.Note that the compositionL∘ϕ of a logharmonic function L with an analytic function ϕ is logharmonic and, also, the composition H∘ϕ of a harmonic function H with analytic function ϕ is harmonic, while this is not true for the function F. Also, if F1(z)=r2L1(z) and F2(z)=r2L2(z) are in LLh(U), where L1 and L2 are logharmonic with respect to the same a, then F1αF2β, α+β=1, is also in LLh(U).Denote the Jacobian ofW by JW. Then(5.29)JW=|Wz|2-|Wz¯|2.
Also let(5.30)λW=|Wz|-|Wz¯|,ΛW=|Wz|+|Wz¯|.
Then JW=λWΛW.


### 5.5.1. The Univalence of Functions with Logharmonic Laplacian

First a lower bound for the area of the range ofF(z)=r2L(z) is established, where L is a starlike univalent logharmonic mapping.Theorem 5.25 (see [47]).
LetF(z)=r2L(z), where L=hg¯ is starlike univalent logharmonic in U, with g(0)=1 and h′(0)=1. Let A(r,F) denote the area of F(Ur), where Ur={z:|z|<r}, for r<1. Then
(5.31)A(r,F)≥2π[-2r+r2-2r33+r42-r55+r66-r88+2ln⁡(1+r)].
Equality holds if and only if L0(z)=r2z(1+z¯/2)/(1+z/2) or one of its rotations.Definition 5.26.
LetL be logharmonic function in U. A complex-valued function of the form F(z)=r2L(z) is starlike in U if it is orientation preserving, F(0)=0,F(z)≠0 when z≠0 and the curve F(reit) is starlike with respect to the origin for each 0<r<1. In other words, ∂argF(reit)/∂t=Re((zFz-z¯Fz¯)/F)>0.Remark 5.27.
Note that starlike functions are univalent inU.The following theorem links starlike functions inLLh(U) with the class of starlike analytic functions.Theorem 5.28 (see [48]).
LetF(z)=r2L(z), where L(z)=h(z)g(z)¯, be a logharmonic function in U with respect to a, where a∈B(U) with a(0)=0. Then F is starlike univalent in U if and only if ψ(z)=h(z)/g(z) is starlike univalent function in U.Corollary 5.29.
The functionr2L(z) is starlike for all conformal starlike functions L.A characterization of the logharmonic Laplacian solutions of the Dirichlet problem in the unit discU is given in [48].Theorem 5.30.
LetF* be an orientation-preserving homeomorphism from ∂U onto ∂U, that is, F*(eit)=eiλ(t), where λ is continuous and strictly monotonically increasing on [0,2π]. Furthermore, suppose that λ(2π)=λ(0)+2π. Then F(z)=z¯|z|2h(z)/h(z)¯ is a univalent solution of the Dirichlet problem in U.For the general caseF(z)=r2L(z)+H(z), a sufficient condition is obtained that makes F locally univalent.Theorem 5.31 (see [48]).
LetF(z)=r2h1(z)g1(z)¯+h2(z)+g2(z)¯ be in the class LLh(U). Suppose that ψ(z)=h1(z)/g1(z) is starlike univalent in U, and |g2′(z)|<|h2′(z)| for z∈U. If
(5.32)Re[g2′(r2h1g1¯)z¯¯]<Re[h2′(r2h1g1¯)z¯],
then JF(z)>0  for z≠0, and F is locally univalent.


### 5.5.2. Landau’s Theorem for Functions with Logharmonic Laplacian

Lewy’s famous theorem [49] states that a harmonic function W is locally univalent in D (univalent in some neighborhood of each point in D) if and only if its Jacobian does not vanish in D.The classical Landau Theorem states that iff is analytic in the unit disc U with f(0)=0,f′(0)=1, and |f(z)|<M for z∈U, then f is univalent in the disc Uρ0={z:|z|<ρ0} with(5.33)ρ0=1M+M2-1,
and f(Uρ0) contains a disc UR0 with R0=Mρ02. This result is sharp, with the extremal function f(z)=Mz(1-Mz)/(M-z) (see [19]).Chen et al. [50] obtained a version of Landau’s Theorem for bounded harmonic mappings of the unit disc. Unfortunately their result is not sharp. Better estimates were given in [51] and later in [52].Specifically, it was shown in [52] that if f is harmonic in the unit disc U with f(0)=0,Jf(0)=1, and |f(z)|<M for z∈U, then f is univalent in the disc Uρ1={z:|z|<ρ1} with(5.34)ρ1=1-22Mπ+8M2,
and f(Uρ1) contains a disc UR1 with R1=π/4M-2M(ρ12/(1-ρ1)). This result is the best known, but not sharp.The following Schwarz lemma for harmonic mappings is due to Grigoryan [52].Lemma 5.32 (Schwarz lemma).
Letf be a harmonic mapping of the unit disc U with f(0)=0 and f(U)⊂U. Then
(5.35)|f(z)|≤4πarctan|z|≤4π|z|,Λf(0)≤4π.Recently Mao et al. [53] established the Schwarz lemma for logharmonic mappings, through which two versions of Landau’s theorem for these mappings were obtained.The next theorem gives Landau’s theorem for functions with logharmonic Laplacian of the formF=r2L(z).Theorem 5.33 (see [47]).
LetL be logharmonic in U such that L(0)=0, JL(0)=1, and |L(z)|<M for z∈U. Then there is a constant 0<ρ2<1 such that F=r2L is univalent in the disc |z|<ρ2, ρ2 is the solution of the equation 1=2ρ2M/(1-ρ22)-2Mρ2/(1-ρ22)2, and f(Uρ2) contains a disc UR2 with R2=ρ22-2Mρ24/(1-ρ22). This result is not sharp.Finally we give a Landau theorem for functions of logharmonic Laplacian of the formF=r2L+K.Theorem 5.34 (see [47]).
LetF=r2L+K,z=reiθ be in LLh(U), where L is logharmonic and K is harmonic in the unit disc U, such that L(0)=K(0)=0,JF(0)=1, and |L| and |K| are both bounded by M. Then there is a constant 0<ρ3<1 such that F is univalent in |z|<ρ3. Specifically ρ3satisfies
(5.36)π4M-2ρ3M-2M(ρ33(1-ρ32)2+1(1-ρ3)2-1)=0,
and F(Uρ3) contains a disc UR3, where
(5.37)R3=π4Mρ3-ρ32M11-ρ32-2Mρ321-ρ3.


## 5.5.1. The Univalence of Functions with Logharmonic Laplacian

First a lower bound for the area of the range ofF(z)=r2L(z) is established, where L is a starlike univalent logharmonic mapping.Theorem 5.25 (see [47]).
LetF(z)=r2L(z), where L=hg¯ is starlike univalent logharmonic in U, with g(0)=1 and h′(0)=1. Let A(r,F) denote the area of F(Ur), where Ur={z:|z|<r}, for r<1. Then
(5.31)A(r,F)≥2π[-2r+r2-2r33+r42-r55+r66-r88+2ln⁡(1+r)].
Equality holds if and only if L0(z)=r2z(1+z¯/2)/(1+z/2) or one of its rotations.Definition 5.26.
LetL be logharmonic function in U. A complex-valued function of the form F(z)=r2L(z) is starlike in U if it is orientation preserving, F(0)=0,F(z)≠0 when z≠0 and the curve F(reit) is starlike with respect to the origin for each 0<r<1. In other words, ∂argF(reit)/∂t=Re((zFz-z¯Fz¯)/F)>0.Remark 5.27.
Note that starlike functions are univalent inU.The following theorem links starlike functions inLLh(U) with the class of starlike analytic functions.Theorem 5.28 (see [48]).
LetF(z)=r2L(z), where L(z)=h(z)g(z)¯, be a logharmonic function in U with respect to a, where a∈B(U) with a(0)=0. Then F is starlike univalent in U if and only if ψ(z)=h(z)/g(z) is starlike univalent function in U.Corollary 5.29.
The functionr2L(z) is starlike for all conformal starlike functions L.A characterization of the logharmonic Laplacian solutions of the Dirichlet problem in the unit discU is given in [48].Theorem 5.30.
LetF* be an orientation-preserving homeomorphism from ∂U onto ∂U, that is, F*(eit)=eiλ(t), where λ is continuous and strictly monotonically increasing on [0,2π]. Furthermore, suppose that λ(2π)=λ(0)+2π. Then F(z)=z¯|z|2h(z)/h(z)¯ is a univalent solution of the Dirichlet problem in U.For the general caseF(z)=r2L(z)+H(z), a sufficient condition is obtained that makes F locally univalent.Theorem 5.31 (see [48]).
LetF(z)=r2h1(z)g1(z)¯+h2(z)+g2(z)¯ be in the class LLh(U). Suppose that ψ(z)=h1(z)/g1(z) is starlike univalent in U, and |g2′(z)|<|h2′(z)| for z∈U. If
(5.32)Re[g2′(r2h1g1¯)z¯¯]<Re[h2′(r2h1g1¯)z¯],
then JF(z)>0  for z≠0, and F is locally univalent.


## 5.5.2. Landau’s Theorem for Functions with Logharmonic Laplacian

Lewy’s famous theorem [49] states that a harmonic function W is locally univalent in D (univalent in some neighborhood of each point in D) if and only if its Jacobian does not vanish in D.The classical Landau Theorem states that iff is analytic in the unit disc U with f(0)=0,f′(0)=1, and |f(z)|<M for z∈U, then f is univalent in the disc Uρ0={z:|z|<ρ0} with(5.33)ρ0=1M+M2-1,
and f(Uρ0) contains a disc UR0 with R0=Mρ02. This result is sharp, with the extremal function f(z)=Mz(1-Mz)/(M-z) (see [19]).Chen et al. [50] obtained a version of Landau’s Theorem for bounded harmonic mappings of the unit disc. Unfortunately their result is not sharp. Better estimates were given in [51] and later in [52].Specifically, it was shown in [52] that if f is harmonic in the unit disc U with f(0)=0,Jf(0)=1, and |f(z)|<M for z∈U, then f is univalent in the disc Uρ1={z:|z|<ρ1} with(5.34)ρ1=1-22Mπ+8M2,
and f(Uρ1) contains a disc UR1 with R1=π/4M-2M(ρ12/(1-ρ1)). This result is the best known, but not sharp.The following Schwarz lemma for harmonic mappings is due to Grigoryan [52].Lemma 5.32 (Schwarz lemma).
Letf be a harmonic mapping of the unit disc U with f(0)=0 and f(U)⊂U. Then
(5.35)|f(z)|≤4πarctan|z|≤4π|z|,Λf(0)≤4π.Recently Mao et al. [53] established the Schwarz lemma for logharmonic mappings, through which two versions of Landau’s theorem for these mappings were obtained.The next theorem gives Landau’s theorem for functions with logharmonic Laplacian of the formF=r2L(z).Theorem 5.33 (see [47]).
LetL be logharmonic in U such that L(0)=0, JL(0)=1, and |L(z)|<M for z∈U. Then there is a constant 0<ρ2<1 such that F=r2L is univalent in the disc |z|<ρ2, ρ2 is the solution of the equation 1=2ρ2M/(1-ρ22)-2Mρ2/(1-ρ22)2, and f(Uρ2) contains a disc UR2 with R2=ρ22-2Mρ24/(1-ρ22). This result is not sharp.Finally we give a Landau theorem for functions of logharmonic Laplacian of the formF=r2L+K.Theorem 5.34 (see [47]).
LetF=r2L+K,z=reiθ be in LLh(U), where L is logharmonic and K is harmonic in the unit disc U, such that L(0)=K(0)=0,JF(0)=1, and |L| and |K| are both bounded by M. Then there is a constant 0<ρ3<1 such that F is univalent in |z|<ρ3. Specifically ρ3satisfies
(5.36)π4M-2ρ3M-2M(ρ33(1-ρ32)2+1(1-ρ3)2-1)=0,
and F(Uρ3) contains a disc UR3, where
(5.37)R3=π4Mρ3-ρ32M11-ρ32-2Mρ321-ρ3.



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*Source: 721943-2012-02-22.xml*