# Fekete-Szegö Inequality for Analytic and Biunivalent Functions Subordinate to Gegenbauer Polynomials

Authors: Ala Amourah; Basem Aref Frasin; Thabet Abdeljawad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the present paper, a subclass of analytic and biunivalent functions by means of Gegenbauer polynomials is introduced. Certain coefficients bound for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szegö problem for this subclass is solved. A number of known or new results are shown to follow upon specializing the parameters involved in our main results.


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## Body

## 1. Introduction

Orthogonal polynomials have been studied extensively as early as they were discovered by Legendre in 1784 [1]. In mathematical treatment of model problems, orthogonal polynomials arise often to find solutions of ordinary differential equations under certain conditions imposed by the model.The importance of the orthogonal polynomials for contemporary mathematics, as well as for a wide range of their applications in physics and engineering, is beyond any doubt. It is well-known that these polynomials play an essential role in problems of the approximation theory. They occur in the theory of differential and integral equations as well as in mathematical statistics. Their applications in quantum mechanics, scattering theory, automatic control, signal analysis, and axially symmetric potential theory are also known [2, 3].Formally speaking, polynomialsPr and Pm of order r and m are orthogonal if
(1)∫abΦxPrxPmxdx=0.forr≠m,where Φx is nonnegative function in the interval a,b; therefore, the integral is well-defined for all finite order polynomials Prx.A special case of orthogonal polynomials is Gegenbauer polynomials. They are representatively related with typically real functionsTR as discovered in [4], where the integral representation of typically real functions and generating function of Gegenbauer polynomials are using common algebraic expressions. Undoubtedly, this led to several useful inequalities appear from the Gegenbauer polynomial realm.Typically, real functions play an important role in the geometric function theory because of the relationTR=co¯SR and its role of estimating coefficient bounds, where SR denotes the class of univalent functions in the unit disk with real coefficients and co¯SR denotes the closed convex hull of SR.This paper associates certain biunivalent functions with Gegenbauer polynomials and then explores some properties of the class in hand. Paving the way for mathematical notations and definitions, we provide the following section.


## 2. Definitions and Preliminaries

LetA denotes the class of all analytic functions f defined in the open unit disk U=z∈ℂ:z<1 and normalized by the conditions f0=0 and f′0=1. Thus, each f∈A has a Taylor-Maclaurin series expansion of the form
(2)fz=z+∑n=2∞anzn,z∈U.Further, letS denotes the class of all functions f∈A which are univalent in U (for details, see [5]).A subordination between two analytic functionsf and g is written as f≺g. Conceptually, the analytic function f is subordinate to g if the image under g contains the image under f. Technically, the analytic function f is subordinate to g if there exists a Schwarz function w with w0=0 and ∣wz∣<1 for all z∈U; such that
(3)fz=gwz.Besides, if the functiong is univalent in U, then the following equivalence holds:
(4)fz≺gzifandonlyiff0=g0fU⊂gU.Further on the subordination principle, we refer to [6].It is well-known that, iffz is an univalent analytic function from a domain D1 onto a domain D2, then the inverse function gz defined by
(5)gfz=z,z∈D1,is an analytic and univalent mapping from D2 to D1. Moreover, by the familiar Koebe one-quarter theorem (for details, see [5]), we know that the image of U under every function f∈S contains a disk of radius 1/4.According to this, every functionf∈S has an inverse map f−1 that satisfies the following conditions:
(6)f−1fz=zz∈U,ff−1w=ww<r0f;r0f≥14.In fact, the inverse function is given by(7)gw=f−1w=w−a2w2+2a22−a3w3−5a23−5a2a3+a4w4+⋯.A functionf∈A is said to be biunivalent in U if both fz and f−1z are univalent in U. Let Σ denotes the class of biunivalent functions in U given by (2). Examples of functions in the class Σ are
(8)z1−z,−log1−z,12log1+z1−z,⋯.It is worth noting that the familiar Koebe function is not a member ofΣ, since it maps the unit disk U univalently onto the entire complex plane except the part of the negative real axis from −1/4 to −∞. Thus, clearly, the image of the domain does not contain the unit disk U. For a brief history and some intriguing examples of functions and characterization of the class Σ, see [7–15].In 1967, Lewin [16] investigated the biunivalent function class Σ and showed that ∣a2∣<1.51. Subsequently, Brannan and Clunie [17] conjectured that ∣a2∣≤2. On the other hand, Netanyahu [18] showed that maxf∈Σ∣a2∣=4/3. The best-known estimate for functions in Σ has been obtained in 1984 by Tan [19], that is, ∣a2∣<1.485. The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients ∣an∣n∈ℕ\1,2 for each f∈Σ given by (2) is presumably still an open problem.The most important and well-investigated subclasses of the analytic and univalent function classS are the class S∗ς of starlike functions of order ς in U and the class Kς of convex functions of order ς in U. By definition, we have
(9)S∗ς≔f:f∈SandRezf′zfz>ς,z∈U;0≤ς<1,Kς≔f:f∈SandRe1+zf′′zf′z>ς,z∈U;0≤ς<1.For0≤ς<1, a function f∈Σ is in the class SΣ∗ς of bistarlike function of order ς or KΣς of biconvex function of order ς if both f and f−1 are, respectively, starlike or convex functions of order ς.Very recently, Amourah [20] considered the Gegenbauer polynomials Hαx,z, which are given by the following recurrence relation:For nonzero real constantα, a generating function of Gegenbauer polynomials is defined by
(10)Hαx,z=11−2xz+z2α,where x∈−1,1 and z∈U. For fixed x, the function Hα is analytic in U, so it can be expanded in a Taylor series as
(11)Hαx,z=∑n=0∞Cnαxzn,where Cnαx is Gegenbauer polynomial of degree n.Obviously,Hα generates nothing when α=0. Therefore, the generating function of the Gegenbauer polynomial is set to be
(12)H0x,z=1−log1−2xz+z2=∑n=0∞Cn0xzn,for α=0. Moreover, it is worth to mention that a normalization of α to be greater than −1/2 is desirable [3, 21]. Gegenbauer polynomials can also be defined by the following recurrence relations:
(13)Cnαx=1n2xn+α−1Cn−1αx−n+2α−2Cn−1αx,with the initial values
(14)C0αx=1,C1αx=2αxandC2αx=2α1+αx2−α.First off, we present some special cases of the polynomialsCnαx:(1)
Forα=1, we get the Chebyshev Polynomials(2)
Forα=1/2, we get the Legendre PolynomialsRecently, many researchers have been exploring biunivalent functions associated with orthogonal polynomials, few to mention [22–28]. For Gegenbauer polynomial, as far as we know, there is little work associated with biunivalent functions in the literatures. Initiating an exploration on the properties of biunivalent functions associated with Gegenbauer polynomials is the main goal of this paper. To do so, we take into account, the following definitions.Definition 1.
Letλ≥1, μ≥0,x∈1/2,1 and α is a nonzero real constant. A function f∈Σ given by (2) is said to be in the class BΣμλ,x,α if the following subordinations are satisfied:
(15)1−λfzzμ+λf′zfzzμ−1≺Hαx,z,(16)1−λf−1wwμ+λf−1w′f−1wwμ−1≺Hαx,w,where the function f−1w is defined by (7) and Hα is the generating function of the Gegenbauer polynomial given by (10).By suitably specializing the parametersμ,λ, and α, the class BΣμλ,x,α leads to the following new subclasses of biuniavlent functions:Example 1.
Ifμ=1 and a function f∈Σ given by (2) is said to be in the class BΣλ,x,α if the following subordinations are satisfied:
(17)1−λfzz+λf′z≺11−2xz+z2α,1−λf−1ww+λf−1w′≺11−2xz+z2α,where x∈1/2,1 and the function f−1w is defined by (7).Example 2.
Ifλ=1 and a function f∈Σ given by (2) is said to be in the class BΣμx,α if the following subordinations are satisfied:
(18)f′zfzzμ−1≺11−2xz+z2α,f−1w′f−1wwμ−1≺11−2xz+z2α,where x∈1/2,1 and the function f−1w is defined by (7).Example 3.
Ifλ=1, μ=1, and a function f∈Σ given by (2) is said to be in the class BΣx,α if the following subordinations are satisfied:
(19)f′z≺11−2xz+z2α,f−1w′≺11−2xz+z2α,where x∈1/2,1 and the function f−1w is defined by (7).Example 4.
Ifλ=1, μ=0 and a function f∈Σ given by (2) is said to be in the class SΣ∗x,α if the following subordinations are satisfied:
(20)zf′zfz≺11−2xz+z2α,zf−1w′f−1w≺11−2xz+z2α,where x∈1/2,1 and the function f−1w is defined by (7).Remark 2.
The subclassesBΣμλ,x,1=BΣμλ,x and BΣ1λ,x,1=BΣλ,x were studied by Bulut et al. [29] and Bulut et al. [30], respectively.In this paper, motivated by recent works of Amourah et al. [20], we use Gegenbauer polynomials to obtain the estimates on the initial Taylor coefficients a2 and a3 for the function class BΣμλ,x,α.Unless otherwise mentioned, we assume in the remainder of this paper thatλ≥1,μ≥0, x∈1/2,1, and α is a nonzero real constant.


## 3. Coefficient Bounds of the ClassBΣμλ,x,α

In the following theorem, we determine the initial Taylor coefficientsa2 and a3 for the function class BΣμλ,x,α.Theorem 3.
Letf∈Σ given by (2) belongs to the class BΣμλ,x,α. Then,
(21)a2≤2αx2αxαμ+λ2−2αμ+λ21+α−α2μ+2λ1+μx2,a3≤4α2x2μ+λ2+2αxμ+2λ.Proof.
Letf∈BΣμλ,x,α. From (15) and (16), we have
(22)1−λfzzμ+λf′zfzzμ−1=1+C1αxc1z+C1αxc2+C2αxc12z2+⋯,(23)1−λf−1wwμ+λf−1w′f−1wwμ−1=1+C1αxd1w+C1αxd2+C2αxd12w2+⋯,for some analytic functions
(24)pz=c1z+c2z2+c3z3+⋯z∈U,qw=d1w+d2w2+d3w3+⋯w∈U,such that p0=q0=0 and ∣pz∣<1,∣qw∣<1 for all z,w∈U.It is fairly well known that ifpz<1,z∈U and qw<1,w∈U, then
(25)cj≤1anddj≤1forallj∈ℕ.Thus, upon comparing the corresponding coefficients in (22) and (23), we have
(26)μ+λa2=C1αxc1,(27)μ+2λμ−12a22+a3=C1αxc2+C2αxc12,(28)−μ+λa2=C1αxd1,(29)μ+2λμ+32a22−a3=C1αxd2+C2αxd12.It follows from (26) and (28) that
(30)c1=−d1,(31)2μ+λ2a22=C1αx2c12+d12.If we add (27) and (29), we get
(32)μ+2λ1+μa22=C1αxc2+d2+C2αxc12+d12.Substituting the value ofc12+d12 from (31) in the right-hand side of (32), we deduce that
(33)μ+2λ1+μ−2μ+λ2C2αxC1αx2a22=C1αxc2+d2.Moreover, computations using (23), (25), and (33), we find that
(34)a2≤2αx2αxαμ+λ2−2αμ+λ21+α−α2μ+2λ1+μx2.Moreover, if we subtract (29) from (27), we obtain
(35)2μ+2λa3−a22=C1αxc2−d2+C2αxc12−d12.Then, in view of (14) and (31), Eq. (35) becomes
(36)a3=C1αx22μ+λ2c12+d12+C1αx2μ+2λc2−d2.Thus, applying (14), we conclude that
(37)a3≤4α2x2μ+λ2+2αxμ+2λ.


## 4. Fekete-Szegö Problem for the Function ClassBΣμλ,x,α

Fekete-Szegö inequality is one of the famous problems related to coefficients of univalent analytic functions. It was first given by [31], who stated that, if f∈Σ, then
(38)a3−ηa22≤1+2e−2η/1−μ.This bound is sharp whenη is real.In this section, we aim to provide Fekete-Szegö inequalities for functions in the classBΣμλ,x,α. These inequalities are given in the following theorem.Theorem 4.
Letf∈Σ given by (2) belongs to the class BΣμλ,x,α. Then,
(39)a3−ηa22≤2αxμ+2λ,η−1≤αμ+λ2−2αμ+λ21+α−α2μ+2λ1+μx24αμ+2λx28αx31−ηαμ+λ2−2αμ+λ21+α−α2μ+2λ1+μx2,η−1≥αμ+λ2−2αμ+λ21+α−α2μ+2λ1+μx24αμ+2λx2.Proof.
From (33) and (35)
(40)a3−ηa22=1−ηC1αx3c2+d2μ+2λ1+μC1αx2−2μ+λ2C2αx+C1αx2μ+2λc2−d2=C1αxhη+12μ+2λc2+hη−12μ+2λd2,where
(41)hη=C1αx21−ημ+2λ1+μC1αx2−2μ+λ2C2αx,Then, in view of (14), we conclude that
(42)a3−ηa22≤C1αxμ+2λ0≤hη≤12μ+2λ,2C1αxhηhη≥12μ+2λ,which completes the proof of Theorem4.


## 5. Corollaries and Consequences

In this section, we apply our main results in order to deduce each of the following new corollaries and consequences.Corollary 5.
Letf∈Σ given by (2) belongs to the class BΣμλ,x,1/2. Then,
(43)a2≤xx1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx2,a3≤x2μ+λ2+xμ+2λ,a3−ηa22≤xμ+2λ,η−1≤1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx22μ+2λx2x31−η1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx2,η−1≥1/2μ+λ2−3/2μ+λ2−1/2μ+2λ1+μx22μ+2λx2.Corollary 6.
Letf∈Σ given by (2) belongs to the class BΣ1λ,x,α=BΣλ,x,α. Then,
(44)a2≤2αx2αxα1+λ2−2α1+λ21+α−2α21+2λx2,a3≤4α2x21+λ2+2αx1+2λ,a3−ηa22≤2αx1+2λ,η−1≤α1+λ2−2α1+λ21+α−2α21+2λx24α1+2λx28αx31−ηα1+λ2−2α1+λ21+α−2α21+2λx2,η−1≥α1+λ2−2α1+λ21+α−2α21+2λx24α1+2λx2.Corollary 7.
Letf∈Σ given by (2) belongs to the class BΣμ1,x,α=BΣμx,α. Then,
(45)a2≤2αx2αxαμ+12−2αμ+121+α−α2μ+21+μx2,a3≤4α2x2μ+12+2αxμ+2,a3−ηa22≤2αxμ+2,η−1≤αμ+12−2αμ+121+α−α2μ+21+μx24αμ+2x28αx31−ηαμ+12−2αμ+121+α−α2μ+21+μx2,η−1≥αμ+12−2αμ+121+α−α2μ+21+μx24αμ+2x2.Corollary 8.
Letf∈Σ given by (2) belongs to the class BΣ01,x,α=SΣ∗x,α. Then,
(46)a2≤2αx2x1−21−αx2,a3≤4α2x2+αx,a3−ηa22≤αx,η−1≤1−21−αx28x28α2x31−η1−21−αx2,η−1≥1−21−αx28x2.Corollary 9.
Letf∈Σ given by (2) belongs to the class BΣ11,x,α=BΣx,α. Then,
(47)a2≤αx2x1−2−αx2,a3≤α2x2+2αx3,a3−ηa22≤2αx3,η−1≤1−2−αx23x22α2x31−η1−2−αx2,η−1≥1−2−αx23x2.


## 6. Concluding Remark

By takingα=1 and specializing μ=1 or λ=1, one can deduce the above results for various subclasses of Σ studied by Bulut et al. [29] and by Altinkaya and Yalcin [32].



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