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
Typically real functions play an important role in the geometric function theory because of the relation
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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