# Differential Subordination Results for Analytic Functions in the Upper Half-Plane

Authors: Huo Tang; M. K. Aouf; Guan-Tie Deng; Shu-Hai Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

There are many articles in the literature dealing with differential subordination problems for analytic functions in the unit disk, and only a few articles deal with the above problems in the upper half-plane. In this paper, we aim to derive several differential subordination results for analytic functions in the upper half-plane by investigating certain suitable classes of admissible functions. Some useful consequences of our main results are also pointed out.


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## Body

## 1. Introduction

LetΔ denote the upper half-plane; that is,
(1)Δ={z∈ℂ:Im⁡(z)>0},
and let ℋ[Δ] denote the class of functions f which are analytic in Δ and which satisfy the so-called hydrodynamic normalization (see [1–3]):
(2)lim⁡Δ∋z→∞[f(z)-z]=0.Also let𝒮[Δ] denote the class of all functions in ℋ[Δ] which are univalent in Δ. Various basic properties concerning functions belonging to the class 𝒮[Δ] were developed in a series of articles (see, for details, [4–6]).A functionf∈ℋ[Δ], with f(z)≠0, is said to be starlike in Δ if and only if
(3)Im⁡{f′(z)f(z)}<0.
We denote by 𝒮*[Δ] the subclass of ℋ[Δ] which consists of functions which are starlike in Δ.A functionf∈ℋ[Δ], with f(z)≠z, is said to be convex in Δ if and only if
(4)Im⁡{f′′(z)f′(z)}>0.Also, we denote by𝒦[Δ] the subclass of ℋ[Δ] which consists of functions which are convex in Δ. The classes 𝒮*[Δ] and 𝒦[Δ] were introduced by Stankiewicz [3].We first need to recall the notion of subordination in the upper half-plane.Letf and g be members of ℋ[Δ]. The function f is subordinate to g, written as f≺g or f(z)≺g(z), if there exists a function φ∈ℋ[Δ] with φ[Δ]⊂Δ such that f(z)=g(φ(z)). Furthermore, if the function g is univalent in Δ, then we have the following equivalence (cf. [7]):
(5)f(z)≺g(z)(z∈Δ)⟺f(Δ)⊂g(Δ).Using methods similar to those used in the unit disk, Răducanu and Pascu [7] have extended the theory of differential subordinations to the upper half-plane. In the following, we will list some definitions and theorems, which are required to prove our main results.Definition 1 (see [8, Definition 8.3i, p.403]).
Denote by𝒬(Δ) the set of functions q∈ℋ[Δ] that are analytic and injective on Δ¯∖E(q), where
(6)E(q)={ξ∈∂Δ:lim⁡z→ξq(z)=∞},
and are such that q′(ξ)≠0 for ξ∈∂Δ∖E(q).Definition 2 (see [7]).
LetΩ be a set in ℂ and q∈𝒬(Δ). The class of admissible functions ΨΔ[Ω,q] consists of those functions ψ:ℂ3×Δ→ℂ that satisfy the following admissibility condition:
(7)ψ(r,s,t;z)∉Ω,
whenever
(8)r=q(ξ),s=kq′(ξ),Im⁡{tq′(ξ)}≥k2Im⁡{q′′(ξ)q′(ξ)},
where z∈Δ, ξ∈∂Δ∖E(q), and k≥0.
Ifψ:ℂ2×Δ→ℂ, then the admissibility condition reduces to
(9)ψ(q(ξ),kq′(ξ);z)∉Ω,
where z∈Δ, ξ∈∂Δ∖E(q), and k≥0.Theorem 3 (see [7]).
Letψ∈ΨΔ[Ω,q] and p∈ℋ[Δ]. If
(10)ψ(p(z),p′(z),p′′(z);z)∈Ω,
for z∈Δ, then
(11)p(z)≺q(z).In the present paper, by making use of the differential subordination results in the upper half-plane of Rǎducanu and Pascu [7] (which is a generalization of results in the unit disk obtained by Miller and Mocanu [8]), we determine certain appropriate classes of admissible functions and investigate some differential subordination properties of analytic functions in the upper half-plane. It should be remarked in passing that, in recent years, several authors obtained many interesting results associated with differential subordination and superordination in the unit disk; the interested reader may refer to, for example, [9–18].


## 2.  The Main Subordination Results

We first define the following class of admissible functions that are required in proving our first result.Definition 4.
LetΩ be a set in ℂ and q∈𝒬(Δ)∩ℋ[Δ]. The class of admissible functions ΦΔ[Ω,q] consists of those functions ϕ:ℂ3×Δ→ℂ that satisfy the following admissibility condition:
(12)ϕ(u,v,w;z)∉Ω,
whenever
(13)u=q(ξ),v=kq′(ξ)q(ξ)(q(ξ)≠0),Im⁡{u(wv+v2)q′(ξ)}≥k2Im⁡{q′′(ξ)q′(ξ)}(z∈Δ;ξ∈∂Δ∖E(q);k≥0).Theorem 5.
Letϕ∈ΦΔ[Ω,q]. If f∈ℋ[Δ] satisfies
(14){ϕ(f′(z)f(z),f′′(z)f′(z)-f′(z)f(z),f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]-f′(z)f(z);zf′(z)f(z)):z∈Δ}⊂Ω,
then
(15)f′(z)f(z)≺q(z)(z∈Δ).Proof.
Define the functionp(z) in Δ by
(16)p(z)=f′(z)f(z).
A simple calculation yields
(17)f′′(z)f′(z)-f′(z)f(z)=p′(z)p(z).
Further computations show that
(18)f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]-f′(z)f(z)=p′′(z)p′(z)-p′(z)p(z).
We now define the transformation from ℂ3 to ℂ by
(19)u(r,s,t)=r,v(r,s,t)=sr,w(r,s,t)=rt-s2rs.
Let(20)ψ(r,s,t;z)=ϕ(u,v,w;z)=ϕ(r,sr,rt-s2rs;z).
Using (16)–(18), and from (20), we obtain
(21)ψ(p(z),p′(z),p′′(z);z)=ϕ(f′(z)f(z),f′′(z)f′(z)-f′(z)f(z),f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]-f′(z)f(z);z).
Hence, (14) becomes
(22)ψ(p(z),p′(z),p′′(z);z)∈Ω.
From (19), we easily get
(23)t=u(wv+v2).
Thus, the admissibility condition for ϕ∈ΦΔ[Ω,q] in Definition 4 is equivalent to the admissibility condition for ψ as given in Definition 2. Therefore ψ∈ΨΔ[Ω,q], and by Theorem 3, we have p(z)≺q(z), or, equivalently, f′(z)/f(z)≺q(z), which evidently completes the proof of Theorem 5.IfΩ≠ℂ is a simply connected domain, then Ω=h(Δ) for some conformal mapping h(z) of Δ onto Ω. In this case, the class ΦΔ[h(Δ),q] is written as ΦΔ[h,q]. The following result is an immediate consequence of Theorem 5.Theorem 6.
Letϕ∈ΦΔ[h,q]. If f∈ℋ[Δ] satisfies
(24)ϕ(f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]f′(z)f(z),f′′(z)f′(z)-f′(z)f(z),f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]-f′(z)f(z);z)≺h(z),
then
(25)f′(z)f(z)≺q(z)(z∈Δ).Our next result is an extension of Theorem5 to the case where the behavior of q(z) on ∂Δ is not known.Theorem 7.
Leth and q be univalent in Δ with q∈𝒬(Δ), and set qρ(z)=q(ρz) and hρ(z)=h(ρz). Let ϕ:ℂ3×Δ→ℂ satisfy one of the following conditions:( 1)
ϕ∈ΦΔ[h,qρ], for some ρ∈(0,1), or(2)
there existsρ0∈(0,1) such that ϕ∈ΦΔ[hρ,qρ] for all ρ∈(ρ0,1).If f∈ℋ[Δ] satisfies (24), then
(26)f′(z)f(z)≺q(z)(z∈Δ).Proof.
The proof of Theorem7 is similar to that of [8, Theorem 2.3d, p.30] and so we choose to omit it.The next theorem yields the best dominant of the differential subordination (24).Theorem 8.
Leth be univalent in Δ and ϕ:ℂ3×Δ→ℂ. Suppose that the following differential equation:
(27)ϕ(q(z),q′(z)q(z),q′′(z)q′(z)-q′(z)q(z);z)=h(z),
has a solution q(z) and satisfies one of the following conditions:(1)
q∈𝒬(Δ) and ϕ∈ΦΔ[h,q],(2)
q is univalent in Δ and ϕ∈ΦΔ[h,qρ], for some ρ∈(0,1), or(3)
q is univalent in Δ and there exists ρ0∈(0,1) such that ϕ∈ΦΔ[hρ,qρ] for all ρ∈(ρ0,1).
Iff∈ℋ[Δ] satisfies (24), then
(28)f′(z)f(z)≺q(z),
and q is the best dominant.Proof.
Following the same arguments as in [8, Theorem 2.3e, p.31], we deduce that q is a dominant from Theorems 6 and 7. Since q satisfies (27), it is also a solution of (24) and therefore q will be dominated by all dominants. Hence, q is the best dominant.In the particular caseq(z)=z, and in view of Definition 4, the class of admissible functions ΦΔ[Ω,q], denoted by ΦΔ[Ω,z], is described below.Definition 9.
LetΩ be a set in ℂ. The class of admissible functions ΦΔ[Ω,z] consists of those functions ϕ:ℂ3×Δ→ℂ such that
(29)ϕ(ξ,kξ,Lξ-k2kξ;z)∉Ω,
whenever z∈Δ, Im⁡(L)=0, ξ∈ℝ∖{0}, and k>0.Corollary 10.
Letϕ∈ΦΔ[Ω,z]. If f∈ℋ[Δ] satisfies(30)ϕ(f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]f′(z)f(z),f′′(z)f′(z)-f′(z)f(z),f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]-f′(z)f(z);z)∈Ω,
then
(31)f′(z)f(z)≺z(z∈Δ).For the special caseΩ=q(Δ)={ω:Im⁡(ω)>0}, the class ΦΔ[Ω,z] is simply denoted by ΦΔ[Δ,z]. Corollary 10 can now be written in the following form.Corollary 11.
Letϕ∈ΦΔ[Δ,z]. If f∈ℋ[Δ] satisfies
(32)Im⁡{ϕ(f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]-f′(z)f(z);z)f′(z)f(z),f′′(z)f′(z)-f′(z)f(z),f(z)[f′′′(z)f′(z)-(f′′(z))2]f′(z)[f(z)f′′(z)-(f′(z))2]-f′(z)f(z);z)}>0,
then
(33)Im⁡{f′(z)f(z)}>0(z∈Δ).Example 12.
Let the functionsA,B:Δ→ℂ be analytic in Δ and satisfy Im⁡A(z)≤0 and Im⁡B(z)≤0. Then, the functions
(34)ϕ1(u,v,w;z)=u+v+A(z),ϕ2(u,v,w;z)=α(u+v)+(1-α)u+B(z)(0≤α≤1)
satisfy the admissibility condition (29) and hence Corollary 10 yields
(35)Im⁡{f′′(z)f′(z)+A(z)}>0⟹Im⁡{f′(z)f(z)}>0,Im⁡{αf′′(z)f′(z)+(1-α)f′(z)f(z)+B(z)}>0⟹Im⁡{f′(z)f(z)}>0.Next, we introduce the following class of admissible functions.Definition 13.
LetΩ be a set in ℂ and q∈𝒬(Δ)∩ℋ[Δ]. The class of admissible functions ΦΔ,1[Ω,q] consists of those functions ϕ:ℂ2×Δ→ℂ that satisfy the admissibility condition
(36)ϕ(q(ξ),q(ξ)+kq′(ξ)q(ξ);z)∉Ω,
where z∈Δ, ξ∈∂Δ∖E(q), and k≥0.Theorem 14.
Letϕ∈ΦΔ,1[Ω,q]. If f∈ℋ[Δ] satisfies
(37){ϕ(f′(z)f(z),f′′(z)f′(z);z):z∈Δ}⊂Ω,
then
(38)f′(z)f(z)≺q(z)(z∈Δ).Proof.
Define the functionp(z) in Δ by
(39)p(z)=f′(z)f(z).
A simple calculation yields
(40)f′′(z)f′(z)=p(z)+p′(z)p(z).
Define the transformation from ℂ2 to ℂ by
(41)u(r,s)=r,v(r,s)=r+sr.
Let(42)ψ(r,s;z)=ϕ(u,v;z)=ϕ(r,r+sr;z).
The proof will make use of Theorem3. Using (39) and (40), and from (42), we get
(43)ψ(p(z),p′(z);z)=ϕ(f′(z)f(z),f′′(z)f′(z);z).
Hence, (37) becomes
(44)ψ(p(z),p′(z);z)∈Ω.
From (42), we see that the admissibility condition for ϕ∈ΦΔ,1[Ω,q] in Definition 13 is equivalent to the admissibility condition for ψ as given in Definition 2. Hence ψ∈ΨΔ[Ω,q], and by Theorem 3, we have p(z)≺q(z) or f′(z)/f(z)≺q(z).We will denote the classΦΔ,1[h(Δ),q] by ΦΔ,1[h,q], where h is the conformal mapping of Δ onto Ω≠ℂ.Theorem 15.
Letϕ∈ΦΔ,1[h,q]. If f∈ℋ[Δ] satisfies
(45)ϕ(f′(z)f(z),f′′(z)f′(z);z)≺h(z),
then
(46)f′(z)f(z)≺q(z).We extend Theorem15 to the case in which the behavior of q(z) on ∂Δ is not known.Theorem 16.
LetΩ⊂ℂ and q be univalent in Δ with q∈𝒬(Δ). Let ϕ∈ΦΔ,1[h,qρ], for some ρ∈(0,1), where qρ(z)=q(ρz). If f∈ℋ[Δ] satisfies (37), then (46) holds.As a special case, whenq(z)=z, we get the following corollary.Corollary 17.
LetΩ be a set in ℂ and let ϕ:ℂ2×Δ→ℂ satisfy
(47)ϕ(ξ,ξ+kξ;z)∉Ω,
whenever z∈Δ, ξ∈ℝ∖{0}, and k≥0. If f∈ℋ[Δ] satisfies
(48)ϕ(f′(z)f(z),f′′(z)f′(z);z)∈Ω,
then
(49)Im⁡{f′(z)f(z)}>0(z∈Δ).In the special caseΩ=q(Δ)={ω:Im⁡(ω)>0}, Corollary 17 reduces to the following corollary.Corollary 18.
Letϕ:ℂ2×Δ→ℂ satisfy
(50)Im⁡{ϕ(ξ,ξ+kξ;z)}≤0,
whenever z∈Δ, ξ∈ℝ∖{0}, and k≥0. If f∈ℋ[Δ] satisfies
(51)Im⁡{ϕ(f′(z)f(z),f′′(z)f′(z);z)}>0,
then
(52)Im⁡{f′(z)f(z)}>0(z∈Δ).Example 19.
Let the functionC:Δ→ℂ be analytic in Δ and satisfy Im⁡C(z)≤0. Then, the function
(53)ϕ(u,v;z)=uv+C(z)
satisfies the admissibility condition (47) and hence Corollary 18 becomes
(54)Im⁡{f′′(z)f(z)+C(z)}>0⟹Im⁡{f′(z)f(z)}>0.



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