# The Generalized Janowski Starlike and Close-to-Starlike Log-Harmonic Mappings

Authors: Maisarah Haji Mohd; Maslina Darus
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

Motivated by the success of the Janowski starlike function, we consider here
closely related functions for log-harmonic mappings of the formf(z)=zh(z)g(z)¯ defined on the open unit disc U. The functions are in the class of the generalized Janowski starlike log-harmonic mapping, Slh*(A,B,α), with the functional zh(z) in the class of the generalized Janowski starlike functions, S*(A,B,α). By means of these functions, we obtained results on the generalized Janowski close-to-starlike log-harmonic mappings, CSTlh(A,B,α).


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## Body

## 1. Introduction

The classS*(A,B) was investigated by Janowski [1] in early 1970, and since then various other subclasses in relation with this Janowski class have been introduced and studied. In that direction, the log-harmonic mappings which have been studied extensively for the past 3 decades, (see [2–10]) were also associated with the Janowski class. Perhaps, the Janowski starlike log-harmonic univalent functions were first introduced by Polatoğlu and Deniz [11].A functionf is said to be log-harmonic on the open unit disc U={z:|z|<1} if it satisfies the nonlinear elliptic partial differential equation: (1.1)f¯z¯f¯=afzf,
where the second dilatation function a∈ℋ(U) (set of all analytic functions defined on U) such that |a(z)|<1 for all z∈U. For analytic functions h and g in U, the function f can be expressed as(1.2)f(z)=h(z)g(z)¯
if f is a nonvanishing log-harmonic mapping and(1.3)f(z)=z|z|2βh(z)g(z)¯
if f vanishes at z=0 but is not identically zero (for Reβ>-1/2, g(0)=1, and h(0)≠0).Letf(z)=zh(z)g(z)¯ be a univalent log-harmonic mapping, where 0∉f(U) or equivalently 0∉hg(U). Then f is starlike log-harmonic mapping if(1.4)Re(zfz-z¯fz¯f)>0.
Results on starlike log-harmonic mapping of order α was given in [6].Motivated by [11], the class of the generalized Janowski log-harmonic starlike functions was introduced in [12]. For real numbers A and B, with -1≤B<A≤1 and 0≤α<1, the family of analytic functions of the form(1.5)p(z)=1+p1z+p2z2+⋯
is in P(A,B,α) if and only if(1.6)p(z)=1+[(1-α)A+αB]ϕ(z)1+Bϕ(z),
where the function ϕ is analytic in U with ϕ(0)=0 and |ϕ(z)|<1. The following lemma is also essential for p(z) to be in P(A,B,α).Lemma 1.1 (see [13]).
The functionp(z)∈P(A,B,α) if and only if
(1.7)|p(z)-1-[(1-α)A+αB]Br21-B2r2|≤(1-α)(A-B)r1-B2r2
for |z|≤r<1.LetS*(A,B,α) denote the class of the generalized Janowski starlike functions of the analytic functions s(z)=z+s2z2+⋯ such that s(z)∈S*(A,B,α) if and only if(1.8)zs′(z)s(z)=p(z)
and p(z)∈P(A,B,α) for z∈U.For univalent log-harmonic mappingf(z)=zh(z)g(z)¯ with g(0)=1 and h(0)≠0, f is in the class of the generalized Janowski starlike log-harmonic mapping denoted by Slh*(A,B,α) if(1.9)|p(z)-1-[(1-α)A+αB]Br21-B2r2|≤(1-α)(A-B)r1-B2r2,
where(1.10)p(z)=h(z)g(z)+zh′(z)g(z)-zg′(z)h(z)h(z)g(z)=1+zh′(z)h(z)-zg′(z)g(z).
Also observe that if f∈Slh*(A,B,α), then(1.11)Re(zfz-z¯fz¯f)≥1-[(1-α)A+αB]1-B.In the present work, we consider the log-harmonic mappingf(z)=zh(z)g(z)¯ in the generalized Janowski starlike functions with the functional zh(z)∈S*(A,B,α). We also study the class of generalized Janowski close-to-starlike in the next section.


## 2. The Generalized Janowski Starlike Log-Harmonic

Theorem 2.1.
Ifzh(z)∈S*(A,B,α), then
(2.1)(1-Br)(1-α)(A-B)/B≤|h(z)|≤(1+Br)(1-α)(A-B)/BforB≠0,e-(1-α)Ar≤|h(z)|≤e(1-α)ArforB=0.Proof.
Sincezh(z)∈S*(A,B,α), Lemma 1.1 yields that for B≠0 we have
(2.2)1-[(1-α)A+αB]r1-Br≤Re(z(zh(z))′zh(z))≤1+[(1-α)A+αB]r1+Br
or
(2.3)-(1-α)(A-B)r1-Br≤Re(zh′(z)h(z))≤(1-α)(A-B)r1+Br.
Simple calculations yield
(2.4)-(1-α)(A-B)-Blog⁡⁡(1-Br)≤log⁡⁡|h(z)|≤(1-α)(A-B)Blog⁡⁡(1+Br),
and the result follows immediately.
ForB=0, Lemma 1.1 yields
(2.5)1-(1-α)Ar≤Re(z(zh(z))′zh(z))≤1+(1-α)Ar,
and the proof is completed similarly.Theorem 2.2.
Letf(z)=zh(z)g(z)¯∈Slh*(A,B,α) with  zh(z)∈S*(A,B,α). Then one has
(2.6)(1-Br)(1-α)(A-B)/B(1+Br)(1-α)(A-B)/B≤|g(z)|≤(1+Br)(1-α)(A-B)/B(1-Br)(1-α)(A-B)/BforB≠0,e-2(1-α)Ar≤|g(z)|≤e2(1-α)ArforB=0.Proof.
It follows from [12] that for f(z)=zh(z)g(z)¯∈Slh*(A,B,α), we have
(2.7)(1-Br)(1-α)(A-B)/B≤|h(z)g(z)|≤(1+Br)(1-α)(A-B)/BforB≠0,e-(1-α)Ar≤|h(z)g(z)|≤e(1-α)ArforB=0.
With these inequalities and Theorem 2.1, we can conclude the following statement.Theorem 2.3.
Letf(z)=zh(z)g(z)¯∈Slh*(A,B,α) with  zh(z)∈S*(A,B,α). Then one has
(2.8)r(1-Br)2(1-α)(A-B)/B(1+Br)(1-α)(A-B)/B≤|f(z)|≤r(1+Br)2(1-α)(A-B)/B(1-Br)(1-α)(A-B)/BforB≠0,re-3(1-α)Ar≤|f(z)|≤re3(1-α)ArforB=0.Proof.
Forf(z)=zh(z)g(z)¯ and |z|=r, it is easy to see that
(2.9)|f(z)|=|zh(z)g(z)¯|=|z||h(z)||g(z)¯|=r|h(z)||g(z)|.
Thus, we can obtain the results from Theorems 2.1 and 2.2.


## 3. The Generalized Janowski Close-to-Starlike Log-Harmonic

LetPlh be  mapping the set of all log-harmonic mappings, and let R be defined on U which are of the form R(z)=K(z)J(z)¯, where K and J are in ℋ(U), K(0)=J(0)=1 and such that ReR(z)>0 for all z∈U. These log-harmonic mappings with positive real part were studied in [5]. Other interesting studies in the same paper were on the close-to starlike log-harmonic mappings. The author then extended the results to close-to starlike of order α log-harmonic mappings [2].In that direction, we say thatF(z)=zH(z)G(z)¯ is the generalized Janowski close-to-starlike log-harmonic mapping if there exist a log-harmonic mapping f(z)=zh(z)g(z)¯∈STlh*(A,B,α) (-1≤B<A≤1 and 0≤α<1), with respect to the second dilatation function a∈ℋ(U) and a log-harmonic mapping with positive real part R∈Plh where its second dilatation function is the same such that(3.1)F(z)=f(z)R(z)
or equivalently(3.2)ReF(z)f(z)>0.
We could also easily derive from (3.1) that(3.3)Re(zFz-z¯Fz¯F)=Re(zfz-z¯fz¯f)+Re(zRz-z¯Rz¯R).The geometrical interpretation is that under a generalized Janowski close-to-starlike log-harmonic mapping, the radius vector of the image of|z|=r<1 never turns back by the amount more than ((1-α)(A-B)/(1-B))π. As special cases, we see that(i)
forα=0 or under the Janowski close-to-starlike log-harmonic mappings, the radius vector of the image of |z|=r<1 never turns back by an amount more than ((A-B)/(1-B))π,(ii)
for whenA=1,  B=-1 or under the close-to-starlike of order α log-harmonic mappings, the radius vector of the image of |z|=r<1 never turns back by an amount more than (1-α)π,(iii)
forα=0,  A=1,  B=-1 or under the close-to-starlike log-harmonic mappings, the radius vector of the image of |z|=r<1 never turns back by an amount more than π.The following theorem gives us the radius of starlikeness forF(z)=zH(z)G(z)¯∈CSTlh(A,B,α).Theorem 3.1.
The radius of starlikeness forF(z)=zH(z)G(z)¯∈CSTlh(A,B,α) is the largest positive root,  r∈(0,1], such that
(3.4)(1-[(1-α)A+αB]r)(1-r)(1+r)-2r(1-Br)>0.Proof.
ForF(z)=zH(z)G(z)¯∈CSTlh(A,B,α), we have
(3.5)Re(zFz-z¯Fz¯F)=Re(zfz-z¯fz¯f)+Re(zRz-z¯Rz¯R),
and since f∈Slh*(A,B,α) and R∈Plh, (3.5) becomes
(3.6)Re(zFz-z¯Fz¯F)≥1-[(1-α)A+αB]r1-Br+-2r1-r2.
Hence,
(3.7)Re(zFz-z¯Fz¯F)>0
if
(3.8)1-[(1-α)A+αB]r1-Br-2r1-r2>0.Corollary 3.2 (see [2]).
The radius of starlikeness forF(z)=zH(z)G(z)¯∈CSTlh is
(3.9)r<2-3.Corollary 3.3 (see [2]).
The radius of starlikeness forF(z)=zH(z)G(z)¯∈CSTlh(α) is
(3.10)r<2-α-α2-2α+31-2α.Corollary 3.4.
The radius of starlikeness forF(z)=zH(z)G(z)¯∈CSTlh(A,B) is the largest positive root,  r∈(0,1], such that
(3.11)(1-Ar)(1-r)(1+r)-2r(1-Br)>0.Proof.
The proof is completed by takingα=0 in (3.4).We need the following theorem from [5] to prove our next result.Theorem
LetR(z)∈Plh, and suppose that a(0)=0. Then, for z∈U, we have
(3.12)e-2|z|/(1-|z|)≤|R(z)|≤e2|z|/(1-|z|).Theorem 3.5.
ForF(z)=zH(z)G(z)¯∈CSTlh(A,B,α) and  f(z)=zh(z)g(z)¯ with  zh(z)∈S*(A,B,α), one has (3.13)r(1-Br)2(1-α)(A-B)/Be-2r/(1-r)(1+Br)(1-α)(A-B)/B≤|F(z)|≤r(1-Br)2(1-α)(A-B)/Be2r/(1-r)(1+Br)(1-α)(A-B)/BforB≠0,re-3(1-α)Ar-(2r/(1-r))≤|F(z)|≤re3(1-α)Ar-(2r/(1-r))forB=0.Proof.
From (3.12) and Theorem 2.3, we have
(3.14)e-2r/(1-r)≤|R(z)|e2r/(1-r),|z|=r<1,r(1-Br)2(1-α)(A-B)/B(1+Br)(1-α)(A-B)/B≤|f(z)|≤r(1+Br)2(1-α)(A-B)/B(1-Br)(1-α)(A-B)/BforB≠0,re-3(1-α)Ar≤|f(z)|≤re3(1-α)ArforB=0,
respectively. Also, we know that for F∈CSTlh(A,B,α), we have F(z)=f(z)R(z) which then leads to the desired result.



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