### Article

Kyungpook Mathematical Journal 2022; 62(2): 257-269

**Published online** June 30, 2022 https://doi.org/10.5666/KMJ.2022.62.2.257

Copyright © Kyungpook Mathematical Journal.

### Coefficient Estimates for a Subclass of Bi-univalent Functions Associated with Symmetric q-derivative Operator by Means of the Gegenbauer Polynomials

Ala Amourah, Basem Aref Frasin^{*}, Tariq Al-Hawary

Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid, Jordan

e-mail : alaammour@yahoo.com

Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan

e-mail : bafrasin@yahoo.com

Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan

e-mail : tariq_amh@bau.edu.jo

**Received**: July 3, 2021; **Revised**: November 5, 2021; **Accepted**: November 16, 2021

### Abstract

In the present paper, a subclass of analytic and bi-univalent functions is defined using a symmetric

**Keywords**: Gegenbauer polynomials, bi-univalent functions, analytic functions, Fekete-Szegö, problem

### 1. Definitions and Preliminaries

Let

Further, let

Let the functions

such that

If the function

and

It is well known that every function

and

where

A function is said to be bi-univalent in

Let

However, the familiar Koebe function is not a member of

are also not members of

Lewin [19] investigated the bi-univalent function class

The coefficient estimate problem for each of the Taylor-Maclaurin coefficients

Similar to the familiar subclasses

Orthogonal polynomials have been studied extensively as early as they were discovered by Legendre in 1784 [18]. 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 the contemporary mathematics, as well as for wide range of their applications in the 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 the mathematical statistics. Their applications in the quantum mechanics, scattering theory, automatic control, signal analysis and axially symmetric potential theory are also known [7, 11].

A special case of orthogonal polynomials are Gegenbauer polynomials. They are representatively related with typically real functions _{R}

Typically real functions play an important role in the geometric function theory because of the relation _{R}_{R}

Very recently, Amourah et al. [4] considered the Gegenbauer polynomials

where

where

Obviously,

for

with the initial values

Special cases of Gegenbauerpolynomials

The theory of

For the convenience, we provide some basic definitions and concept details of

**Definition 1.1.** ([15]) For

From (1.8), we have

where

is sometimes called

For a function

and

where

**Definition 1.2.** ([8]) The symmetric

From (1.11), we deduce that

when

Clearly, we have the following relations

and

From (1.2) and (1.11), we also deduce that

Recently, many researchers have been exploring bi-univalent functions associated with orthogonal polynomials, few to mention (see,[31], [30]). For Gegenbauer polynomial, as far as we know, there is little work associated with bi-univalent functions in the literatures. Inspired by the works of Amourah et al. [4], we introduce the following new subclasses of bi-univalent functions, as follows:

**Definition 1.3.** Let α is a nonzero real constant. A function

and

where

We note that

**Definition 1.4.** Let α is a nonzero real constant. A function

and

where

**Remark 1.5.** We note that the subclasses

The following result will be required for proving our results.

**Lemma 1.6.** ([23]) Let

then

In this paper, we use the Gegenbauer polynomial expansions to provide estimates for the initial coefficients of the subclass of bi-univalent functions

Unless otherwise mentioned, we assume in the reminder of this paper that,

### 2. Coefficient Bounds of the Class B ˜ Σ q ( x , α )

This section isdevoted to find initial coefficient bounds of the class

**Theorem 2.1.** Let

and

and

Next, define the functions

and

In the following, one can derive

and

From the equalities (2.1), (2.2), (2.3) and (2.4), we obtain that

and

Thus, upon comparing the corresponding coefficients in (2.5) and (2.6), we have

and

It follows from (2.7) and (2.9) that

and

If we add (2.8) and (2.10), we get

Substituting the value of

Using (2.6), (1.16) and (2.14), we find that

Moreover, if we subtract (2.10) from (2.8), we obtain

Then, in view of (1.7) and (2.12), equation (2.15) becomes

Thus applying (1.7) and (1.16), we have

### 3. Fekete-Szegö Problem for the Function Class B ˜ Σ q ( x , α )

Fekete-Szegö inequality is one of the famous problem related to coefficients of univalent analytic functions. It was first given by [12], who stated that, if

This bound is sharp when η is real.

In this section, we aim to provide Fekete-Szegö inequalities for functions in the class

**Theorem 3.1.** Let

where

where

Then, in view of (1.7) and (1.16), we conclude that

Which completes the proof of Theorem 3.1.

### 4. Corollaries and Consequences

In this section, we apply our main results in order to deduce each of the following new corollaries and consequences.

**Corollary 4.1.** Let

and

where

**Corollary 4.2.** Let

and

where

### Concluding Remark.

By taking

### Acknowledgements

The authors would like to thank the referees for their helpful comments and suggestions.

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