# Coefficient Estimates for Two New Subclasses of Biunivalent Functions with respect to Symmetric Points

Authors: Şahsene Altınkaya; Sibel Yalçın
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We introduce two subclasses of biunivalent
functions and find estimates on the coefficientsa2 and a3 for functions in these new subclasses. Also, consequences of the results are pointed out.


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## Body

## 1. Introduction and Definitions

LetA denote the class of analytic functions in the unit disk
(1)U=z∈C:z<1
that have the form
(2)f(z)=z+∑‍n=2∞anzn.
Further, by S we will denote the class of all functions in A which are univalent in U.The Koebe one-quarter theorem [1] states that the image of U under every function f from S contains a disk of radius 1/4. Thus every such univalent function has an inverse f-1 which satisfies
(3)f-1fz=z,z∈U,ff-1w=w,w<r0f,r0f≥14,
where
(4)f-1w=w-a2w2+2a22-a3w3-5a23-5a2a3+a4w4+⋯.A functionf(z)∈A is said to be biunivalent in U if both f(z) and f-1(z) are univalent in U. Let Σ denote the class of biunivalent functions defined in the unit disk U.If the functionsf and g are analytic in U, then f is said to be subordinate to g, written as
(5)fz≺gz,z∈U
if there exists a Schwarz function wz, analytic in U, with
(6)w0=0,wz<1z∈U
such that
(7)fz=gwzz∈U.Lewin [2] studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient a2. Subsequently, Netanyahu [3] showed that max⁡a2=4/3 if f(z)∈Σ. Brannan and Clunie [4] conjectured that a2≤2 for f∈Σ. Brannan and Taha [5] introduced certain subclasses of the biunivalent function class Σ similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions. They introduced bistarlike functions and obtained estimates on the initial coefficients. Bounds for the initial coefficients of several classes of functions were also investigated in [6–15].Not much is known about the bounds on the general coefficientan for n≥4. In the literature, there are only a few works determining the general coefficient bounds an for the analytic biunivalent functions ([16–20]). The coefficient estimate problem for each of an(n∈N∖{1,2};N={1,2,3,…}) is still an open problem.ByS*ϕ and Cϕ we denote the following classes of functions:
(8)S*ϕ=f:f∈A,zf′zfz≺ϕz;z∈U,Cϕ=f:f∈A,1+zf′′zf′z≺ϕz;z∈U.The classesS*ϕ and Cϕ are the extensions of classical sets of starlike and convex functions and in such form were defined and studied by Ma and Minda [21].In [22], Sakaguchi introduced the class SS* of starlike functions with respect to symmetric points in U, consisting of functions f∈A that satisfy the condition Re(zf′(z)/(f(z)-f(-z)))>0, z∈U. Similarly, in [23], Wang et al. introduced the class CS of convex functions with respect to symmetric points in U, consisting of functions f∈A that satisfy the condition Re(zf′(z)′/(f′(z)+f′(-z)))>0, z∈U. In the style of Ma and Minda, Ravichandran (see [24]) defined the classes SS*ϕ and CSϕ.A functionf∈A is in the class SS*ϕ if
(9)2zf′zfz-f-z≺ϕz,z∈U,
and in the class CSϕ if
(10)2zf′z′f′z+f′-z≺ϕz,z∈U.In this paper, we introduce two new subclasses of biunivalent functions. Further, we find estimates on the coefficientsa2 and a3 for functions in these subclasses.


## 2. Coefficient Estimates for the Function ClassSS,Σ*(α,h)

Definition 1.
Let the functionsh,p:U→C be so constrained that
(11)min⁡Rehz,Repz>0,h0=p0=1.Definition 2.
A functionf∈Σ is said to be in the class SS,Σ*(α,h) if the following conditions are satisfied:
(12)21-αzf′z+αzzf′z′1-αfz-f-z+αzf′z+f′-z∈hU,21-αwg′w+αwwg′w′1-αgw-g-w+αwg′w+g′-w∈pU,
where g(w)=f-1(w).Definition 3.
One notes that, forα=0, one gets the class SS*(h) which is defined as follows:
(13)2zf′zfz-f-z∈hU,2wg′wgw-g-w∈pU.Theorem 4.
Letf given by (2) be in the class SS,Σ*(α,h). Then
(14)a2≤min⁡h′02+p′0281+α2,h′′0+p′′081+2α,a3≤min⁡h′02+p′0281+α2+h′′0+p′′081+2α,h′′041+2αh′02+p′0281+α2.Proof.
Letf∈SS,Σ*(α,h) and g be the analytic extension of f-1 to U. It follows from (12) that
(15)2zf′z+αz2f′′z1-αfz-f-z+αzf′z+f′-z=hz,z∈U,2wg′w+αw2g′′w1-αgw-g-w+αwg′w+g′-w=pw,w∈U,
where h(z) and p(w) satisfy the conditions of Definition 1. Furthermore, the functions h(z) and p(w) have the following Taylor-Maclaurin series expansions:(16)hz=1+h1z+h2z2+⋯,(17)pw=1+p1w+p2w2+⋯,
respectively. From (15), we deduce
(18)21+αa2=h1,(19)21+2αa3=h2,(20)-21+αa2=p1,(21)21+2α2a22-a3=p2.
From (18) and (20) we obtain
(22)h1=-p1,(23)81+α2a22=h12+p12.
By adding (19) to (21), we get
(24)41+2αa22=h2+p2.
Therefore, we find from (23) and (24) that
(25)a22≤h′02+p′0281+α2,a22≤h′′0+p′′081+2α.
Subtracting (21) from (19) we have
(26)41+2αa3-41+2αa22=h2-p2.
Then, upon substituting the value of a22 from (23) and (24) into (26), it follows that
(27)a3=h12+p1281+α2+h2-p241+2α,a3=h2+p241+2α+h2-p241+2α.
We thus find that
(28)a3≤h′02+p′0281+α2+h′′0+p′′081+2α,a3≤h′′041+2α.
This completes the proof of Theorem 4.Takingα=0 we get the following.Corollary 5.
Iff∈SS*h then
(29)a2≤min⁡h′02+p′028,h′′0+p′′08,a3≤min⁡h′02+p′028+h′′0+p′′08,h′′04h′02+p′028.Corollary 6.
If we let(30)ϕz=1+z1-zβ=1+2βz+2β2z2+⋯0<β≤1,
then inequalities (14) become
(31)a2≤min⁡β1+α,β1+2α,a3≤min⁡β21+α2+β21+2α,β21+2α.Corollary 7.
If we let(32)ϕz=1+1-2βz1-z=1+21-βz+21-βz2+⋯0≤β<1,
then inequalities (14) become
(33)a2≤min⁡1-β1+α,1-β1+2α,a3≤min⁡1-β21+α2+1-β1+2α,1-β1+2α.Remark 8.
Corollaries6 and 7 provide an improvement of the estimate a3 obtained by Altınkaya and Yalçın [25].Remark 9.
The estimates on the coefficientsa2 and a3 of Corollaries 6 and 7 are improvement of the estimates in [7].


## 3. Coefficient Estimates for the Function Class£S,Σ(α,h)

Definition 10.
A functionf∈Σ is said to be £S,Σ(α,h) if the following conditions are satisfied:
(34)2zf′zfz-f-zα2zf′z′f′z+f′-z1-α∈hz,2wg′wgw-g-wα2wg′w′g′w+g′-w1-α∈pw,
where g(w)=f-1(w).We note that, forα=1, the class £S,Σ(α,h) reduces to the class SS*(h).Definition 11.
One notes that, forα=0, one gets the class CS(h) which is defined as follows:
(35)2zf′z′f′z+f′-z∈hU,2wg′w′g′w+g′-w∈pU.Theorem 12.
Letf given by (2) be in the class £S,Σ(α,h). Then
(36)a2≤min⁡h′02+p′0282-α2,h′′0+p′′083-3α+α2,(37)a3≤min⁡h′02+p′0282-α2+h′′0+p′′083-2α,6-5α+α2h′′0+α-α2p′′083-2α3-3α+α2h′02+p′0282-α2+h′′0+p′′083-2α.Proof.
Letf∈£S,Σ(α,h) and g be the analytic extension of f-1 to U. We have
(38)2zf′zfz-f-zα2zf′z′f′z+f′-z1-α=1+22-αa2z+23-2αa3-2α1-αa22z2+⋯,2wg′wgw-g-wα2wg′w′g′w+g′-w1-α=1-22-αa2w+23-2α2a22-a3-2α1-αa22w2+⋯.
It follows from (34) that
(39)2zf′zfz-f-zα2zf′z′f′z+f′-z1-α=hz,z∈U,2wg′wgw-g-wα2wg′w′g′w+g′-w1-α=pw,w∈U,
where h(z) and p(w) satisfy the conditions of Definition 1.
From (39), we deduce
(40)22-αa2=h1(41)23-2αa3-2α1-αa22=h2,(42)-22-αa2=p1(43)23-2α2a22-a3-2α1-αa22=p2.
From (40) and (42) we obtain
(44)h1=-p1,(45)82-α2a22=h12+p12.
By adding (41) to (43), we get
(46)43-3α+α2a22=h2+p2,
which gives us the desired estimate on a2 as asserted in (36).
Subtracting (43) from (41) we have
(47)43-2αa3-43-2αa22=h2-p2.
Then, in view of (45) and (46), it follows that
(48)a3=h12+p1282-α2+h2-p243-2α,a3=h2+p243-3α+α2+h2-p243-2α
as claimed. This completes the proof of Theorem 12.Takingα=0 we get the following.Corollary 13.
Iff∈CS(h) then
(49)a2≤min⁡h′02+p′0232,h′′0+p′′024,(50)a3≤min⁡h′02+p′0232+h′′0+p′′024,h′′012h′02+p′0232.Corollary 14.
If we let(51)ϕz=1+z1-zβ=1+2βz+2β2z2+⋯0<β≤1,
then inequalities (36) and (37) become
(52)a2≤min⁡β2-α,β3-3α+α2,a3≤min⁡β22-α2+β23-2α,β23-3α+α2.Corollary 15.
If we let(53)ϕz=1+1-2βz1-z=1+21-βz+21-βz2+⋯0≤β<1,
then inequalities (36) and (37) become
(54)a2≤min⁡1-β2-α,1-β3-3α+α2,a3≤min⁡1-β22-α2+1-β3-2α,1-β3-3α+α2.Remark 16.
Corollaries14 and 15 provide an improvement of the estimate a3 obtained by Altınkaya and Yalçın [25].Remark 17.
The estimates on the coefficientsa2 and a3 of Corollaries 14 and 15 are improvement of the estimates obtained in [7].



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