# Fekete-Szegö Functional for Bi-univalent Functions Related with Gegenbauer Polynomials

Authors: Ibrar Ahmad; Syed Ghoos Ali Shah; Saqib Hussain; Maslina Darus; Babar Ahmad
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we introduce and investigate a new subclass of bi-univalent functions related with generating function of Gegenbauer polynomials. We will mainly find bounds on Maclaurin series coefficients for functions belonging to this class. We also study the famous Fekete-Szegö type problem for this subclass which is obtained. We will also point out many special cases as corollaries.


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## Body

## 1. Introduction

LetA denote the class of all functions f analytic in the open unit disk U=z∈ℂ:z<1 which are normalized by the conditions f0=0 and f′0=1. So, each f∈A has a Taylor’s series expansion of the form:(1)fz=z+∑n=2∞anzn.A functionf∈A is said to be univalent in U if it never takes the same value twice. Further, by S, we mean the subclass of A containing all univalent functions, see [1]. If the functions f and F are analytic in U, then f is said to subordinate to F (denoted by f≺F) if there exists a Schwarz function w analytic in U with w0=0 and wz<1, such that(2)fz=Fwz.Further, ifFz is univalent in U, then see [2],(3)fz≺Fz⇔f0=F0,fU⊂FU,z∈U.The Koebe one-quarter theorem (for detail, see [1]) ensures that the image of U under every univalent function f∈A contains a disc of radius 1/4.According to this, every univalent functionf∈A has an inverse map f−1 that satisfies the following conditions:(4)f−1fz=zz∈U,ff−1w=ww<r0f:r0f≥14.The series representation of an inverse function is(5)gw=f−1w=w−b0w2+2b02−b1w3−5b03−5b0b1+b4w4+⋯.A functionf∈A is said to be bi-univalent in U if both f and f−1 are univalent in U. Let Σ denote the class of bi-univalent functions in U.The study of bi-univalent functions was initiated by Lewin [3] and he has shown that b0<1.51. The study of bi-univalent functions is one of the hot areas of ongoing research in the field of geometric function theory. Many subclasses of bi-univalent functions were explored by several authors and they estimated nonsharp bounds of b0 and b1. A brief history of bi-univalent functions can be found in classical articles [4–11] (for more detailed in-depth study, refer [2, 3, 12–15]).Two well-known normalized subclasses of univalent functions with some nice geometrical interpretations are starlike and convex functions. A natural generalization of the class of starlike functions is the classS∗ζ, defined as(6)S∗ζ=f:f∈SandRezf′zfz>ζ,z∈U;0≤ζ<1.Similarly, the class of convex functionskζ is given by(7)kζ=f:f∈SandRe1+zf″zf′z>ζ,z∈U;0≤ζ<1.For0≤ζ<1, a function f∈Σ is in the class SΣ∗ζ of the bistarlike functions of order ζ or (kΣζ of the biconvex function of order ζ) if both f and f−1 in class SΣ∗ζ or kΣζ. The purpose of present investigation is to derive various properties of bi-univalent functions using the famous Gegenbauer polynomials Cnαx which was very recently introduced by Amourah [16] (for details, see [17–22]).For nonzero real constantα, a generating function of Gegenbauer polynomials is given by(8)Hαx,z=11−2xz+z2α,where x∈−1,1. For fixed x, the function Hαx,z is analytic in U, and it can be expanded as(9)Hαx,z=∑n=0∞Cnαxzn,where Cnαx is the Gegenbauer polynomials of degree n. For α=0, the generating function of the Gegenbauer polynomial can also be represented as(10)Hαx,z=1−log1−2x+z2=∑n=0∞Cn0xzn.It can be easily seen that the Gegenbauer polynomials satisfies the following recursive relation:(11)Cnαx=1n2xn+α−1Cn−1αx−n+2α−1Cn−1αx.The first three Gegenbauer polynomials are given as follows:(12)C0αx=1,C1αx=2αx,C2αx=2α1+αx2−α.By takingα=1/2 and α=1, Gegenbauer polynomials reduce to Legendre and Chebyshev polynomials.Recently, researchers explored bi-univalent functions in connection with orthogonal polynomials (see [16, 23–27]). It is important to mention that a very little is known about Gegenbauer polynomials in the context of bi-univalent functions. By taking motivation from about cited work and application of Gegenbauer polynomials, we introduce the following class of bi-univalent functions.Definition 1.
Letλ≥1,μ≥0,δ≥0,1/2<x<1, and α is a nonzero real constant. Then, a function f∈Σ given by (1) is in class βΣμλ,x,α,δ if it satisfied(13)1−λfzzμ+λf′zfzzμ−1+ξδzf″z≺Hαx,z,and(14)1−λgwwμ+λg′wgwwμ−1+ξδwg″w≺Hαx,w.where the function gw is given by (5) and ξ=2λ+μ/2λ+1.
By specializing the parametersμ,λ,α, and δ, the class βΣμλ,x,α,δ leads to the following interesting case.
Special cases:(i)
Letδ=0 and f∈Σ is given by (1). Then, f∈βΣμλ,x,α, if the following subordinations are satisfied:(15)1−λfzzμ+λf′zfzzμ−1≺Hαx,z,1−λgwwμ+λg′wgwwμ−1≺Hαx,w,where1/2<x<1, and the function gw is defined by (5).(ii)
Letδ=0 and μ=1 and a function f∈Σ given by (1). Then, f∈βΣλ,x,α, if the following subordinations are satisfied:(16)1−λfzz+λf′z≺11−2xz+z2α,1−λgww+λg′w≺11−2xw+w2α,where1/2<x<1, and the function gw is defined by (5).(iii)
Letδ=0 and λ=1 and a function f∈Σ given by (1). Then, f∈βΣμx,α if the following subordinations are satisfied:(17)f′zfzzμ−1≺11−2xz+z2α,g′wgwwμ−1≺11−2xw+w2α,where1/2<x<1, and the function gw is defined by (5).(iv)
Letδ=0, λ=1, and μ=1, and a function f∈Σ given by (1). Then, f∈βΣx,α if the following subordinations are satisfied:(18)f′z≺11−2xz+z2α,g′w≺11−2xw+w2α,where1/2<x<1, and the function gw is defined by (5).(v)
Letδ=0, λ=1, and μ=0, and a function f∈Σ given by (1). Then, f∈SΣ∗x,α if the following subordinations are satisfied:(19)zf′zfz≺11−2xz+z2α,wg′wgw≺11−2xw+w2α,where1/2<x<1, and the function gw is defined by (5).Remark 1.
The subclassesβΣμλ,x,α,0=βΣμλ,x,α,βΣμλ,x,1,0=βΣμλ,x, and βΣ1λ,x,1,0=βΣλ,x were studied respectively by Amourah et al. [16] and Bulut et al. (see [28, 29]), see [23, 26, 30].
To avoid repetitions, through out in this paper, we shall assumeλ≥1, μ≥0, δ≥0, 1/2<x<1, and α is a nonzero real constant.


## 2. Coefficient Bounds for the ClassβΣμλ,x,α,δ

In the following theorem, we determine the initial Taylor coefficientsb0 and b1 for the function class βΣμλ,x,α,δ.Theorem 1.
Letf∈βΣμλ,x,μ,δ be given by (1). Then,(20)b0≤2αx2αxβ1,and(21)b1≤2αxμ+2λ+6ξδ+2x2α2β1,where(22)β1=αμ+λ+2ξδ2−2α1+αμ+λ+2ξδ2−α2μ+2λμ+1+12ξδx2.Proof.
Letf∈βΣμλ,x,μ. Then, from (13), we have(23)1−λfzzμ+λf′zfzzμ−1+ξδzf″z=1+C1αxc1z+C1αxc2+C2αxc12z2+⋯.
Also from (14), we have(24)1−λgwwμ+λg′wgwwμ−1+ξδwg″w=1+C1αxd1w+C1αxd2+C2αxd12w2+⋯.
Let(25)pz=c1z+c2z2+c3z3+⋯z∈U,qw=d1w+d2w2+d3w3+⋯w∈U.
Withpz<1 and qw<1, for all z,w∈U.
It is fairly well-known that ifpz<1 and qw<1, then(26)cj≤1,dj≤1,for allj∈N.
Comparing the corresponding coefficients in (24) and (28), we have(27)μ+λ+2ξδb0=C1αxc1,(28)μ+2λμ−12b02+1+6δ1+2λb1=C1αxc2+C2αxc12,(29)−μ+λ+2ξδb0=C1αxd1,(30)μ+2λμ+32+12δ1+2λb0−1+6δ1+2λb1=C1αxd2+C2αxd12.
It follows from (27) and (29) that(31)c1=−d1,(32)2μ+λ+2ξδ2b02=C1αx2c12+d12.
By adding (28) and (30), we obtain(33)μ+2λμ+1+12δ1+2λb02=C1αxc2+d2+C2αxc12+d12.
Substituting the value ofc12+d12 from (32) in the right-hand side of equation (33), we obtain(34)μ+2λμ+1+12ξδ−2C2αxC1αx2μ+λ+2ξδ2b02=C1αxc2+d2.
By using (12), (26), and (34), we obtain (20).
Moreover, if we subtract (30) from (28), we have(35)2μ+2λ1+6δ1+2λb1−2μ+2λ1+6δ1+2λb0=C1αxc2−d2+C2αxc12−d12.
Further, in view of (31), it follows from (35) that(36)b1=b0+C1αxc2−d22μ+2λ+6ξδ.
Hence, applying (12), we will get (21).


## 3. Fekete-Szegö Problem for the Function ClassβΣμλ,x,α,δ

Recall that, forf∈Σ, in [31], Fekete-Szegö showed that(37)b1−ηb02≤1+2e−2η/1−μ.The inequalities (37) are known as Fekete-Szegö problem.In this section, our aim is to study Fekete-Szegö type inequality for the function in the classβΣμλ,x,α,δ.Theorem 2.
Letf∈βΣμλ,x,α,δ be given by (1). Then,(38)b1−ηb02≤2αxμ+2λ+6ξδ,η−1≤β14αμ+2λ+6ξδx2,2x2α21−ηβ1,η−1≥β14αμ+2λ+6ξδx2,where β1 is given by (22).Proof.
From (34) and (35), it follows that(39)b1−ηb02=C1αxc2−d22μ+2λ+6ξδ+1−ηC1αx3c2+d2ξ,=C1αxhη+12μ+2λ+6ξδc2+hη−12μ+2λ+6ξδd2,where(40)hη=C1αx21−ηξ.
Also,(41)ξ=C1αx2μ+2λμ+1+12ξδ−2C2αxμ+2λ+2ξδ2.
In view of (12), we conclude that(42)b1−ηb02≤2xα2μ+2λ+6ξδ,hη≤12μ+2λ+6ξδ,4xαhη,hη≥12μ+2λ+6ξδ,where hη is defined by (40).


## 4. Corollaries

In this section, we apply our main results in order to deduce the following corollaries and consequences.Corollary 1.
If we takeδ=0, in Theorems (1) and (2), then(43)b0≤2αx2xααμ+λ2−2α1+αμ+λ2−α2μ+2λ1+μx2,b1≤2xα1μ+2λ+2xαμ+λ2,b1−ηb02≤2αxμ+2λ,η−1≤μ+λ2−21+αμ+λ2−αμ+2λ1+μx24μ+2λx2,2x3α21−ημ+λ2−21+αμ+λ2−αμ+2λ1+μx2,η−1≥μ+λ2−21+αμ+λ2−αμ+2λ1+μx24μ+2λx2.Corollary 2.
If we takeδ=0 and α=1/2, in Theorems (1) and (2), then(44)b0≤xx1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx2,b1≤xμ+2λ+x2μ+λ2,b1−ηb02≤xμ+2λ,η−1≤1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx22μ+2λx2,x31−η1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx2,η−1≥1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx22μ+2λx2.Corollary 3.
If we takeδ=0 and μ=1, in Theorems (1) and (2), then(45)b0≤2xα31+λ2−21+α1+λ2−2α1+2λx2,b1≤2xα11+2λ+2xα1+λ2,b1−ηb02≤2αx1+2λ,η−1≤1+λ2−21+α1+λ2−2α1+2λx241+2λx2,2x3α21−η1+λ2−21+α1+λ2−2αμ+2λx2,η−1≥1+λ2−21+α1+λ2−2α1+2λx241+2λx2.Corollary 4.
If we takeδ=0 and λ=1, in Theorems (1) and (2), then(46)b0≤2xα3μ+12−21+αμ+12−αμ+21+μx2,b1≤2xα1μ+2+2xαμ+12,b1−ηb02≤2αxμ+2,η−1≤μ+12−21+αμ+12−αμ+21+μx24μ+2λx2,2x3α21−ημ+12−21+αμ+12−αμ+21+μx2,η−1≥μ+12−21+αμ+12−αμ+21+μx24μ+2λx2.Corollary 5.
If we takeδ=0, μ=1, and λ=1, in Theorems (1) and (2), then(47)b0≤2xα2xα1−21−αx2,b1≤αx1+4αx,b1−ηb02≤αx,η−1≤1−21−αx28x2,8α2x31−η1−21−αx2,η−1≥1−21−αx28x2.Corollary 6.
If we takeδ=0, μ=1, and λ=1, in Theorems (1) and (2), then(48)b0≤αx2x1−2−αx2,b1≤αx23+αx,b1−ηb02≤2αx3,η−1≤1−2−αx23x2,2α2x31−η1−2−αx2,η−1≥1−2−αx23x2.


## 5. Conclusion

In this paper, we have introduced and investigated the Taylor-Maclaurin coefficient estimates for a new subclass of analytic and bi-univalent functions defined in the open unit disc. We have also studied Fekete-Szegö type inequality and deduced several corollaries and special cases of each of our main results [32].



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