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.
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, polynomials and of order and are orthogonal if where is nonnegative function in the interval ; therefore, the integral is well-defined for all finite order polynomials .
A special case of orthogonal polynomials is Gegenbauer polynomials. They are representatively related with typically real functions 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 relation and its role of estimating coefficient bounds, where denotes the class of univalent functions in the unit disk with real coefficients and denotes the closed convex hull of .
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
Let denotes the class of all analytic functions defined in the open unit disk and normalized by the conditions and . Thus, each has a Taylor-Maclaurin series expansion of the form
Further, let denotes the class of all functions which are univalent in (for details, see [5]).
A subordination between two analytic functions and is written as . Conceptually, the analytic function is subordinate to if the image under contains the image under . Technically, the analytic function is subordinate to if there exists a Schwarz function with and for all ; such that
Besides, if the function is univalent in , then the following equivalence holds:
Further on the subordination principle, we refer to [6].
It is well-known that, if is an univalent analytic function from a domain onto a domain , then the inverse function defined by is an analytic and univalent mapping from to . Moreover, by the familiar Koebe one-quarter theorem (for details, see [5]), we know that the image of under every function contains a disk of radius .
According to this, every function has an inverse map that satisfies the following conditions:
In fact, the inverse function is given by
A function is said to be biunivalent in if both and are univalent in . Let denotes the class of biunivalent functions in given by (2). Examples of functions in the class are
It is worth noting that the familiar Koebe function is not a member of , since it maps the unit disk univalently onto the entire complex plane except the part of the negative real axis from to . Thus, clearly, the image of the domain does not contain the unit disk . For a brief history and some intriguing examples of functions and characterization of the class , see [7–15].
In 1967, Lewin [16] investigated the biunivalent function class and showed that . Subsequently, Brannan and Clunie [17] conjectured that On the other hand, Netanyahu [18] showed that The best-known estimate for functions in has been obtained in 1984 by Tan [19], that is, . The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients for each given by (2) is presumably still an open problem.
The most important and well-investigated subclasses of the analytic and univalent function class are the class of starlike functions of order in and the class of convex functions of order in . By definition, we have
For , a function is in the class of bistarlike function of order or of biconvex function of order if both and are, respectively, starlike or convex functions of order
Very recently, Amourah [20] considered the Gegenbauer polynomials , which are given by the following recurrence relation:
For nonzero real constant , a generating function of Gegenbauer polynomials is defined by where and . For fixed , the function is analytic in , so it can be expanded in a Taylor series as where is Gegenbauer polynomial of degree .
Obviously, generates nothing when . Therefore, the generating function of the Gegenbauer polynomial is set to be for . Moreover, it is worth to mention that a normalization of to be greater than is desirable [3, 21]. Gegenbauer polynomials can also be defined by the following recurrence relations: with the initial values
First off, we present some special cases of the polynomials (1)For we get the Chebyshev Polynomials(2)For we get the Legendre Polynomials
Recently, 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. Let, and is a nonzero real constant. A function given by (2) is said to be in the class if the following subordinations are satisfied: where the function is defined by (7) and is the generating function of the Gegenbauer polynomial given by (10).
By suitably specializing the parameters , and , the class leads to the following new subclasses of biuniavlent functions:
Example 1. If and a function given by (2) is said to be in the class if the following subordinations are satisfied: where and the function is defined by (7).
Example 2. If and a function given by (2) is said to be in the class if the following subordinations are satisfied: where and the function is defined by (7).
Example 3. If , , and a function given by (2) is said to be in the class if the following subordinations are satisfied: where and the function is defined by (7).
Example 4. If , and a function given by (2) is said to be in the class if the following subordinations are satisfied: where and the function is defined by (7).
Remark 2. The subclasses and 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 and for the function class
Unless otherwise mentioned, we assume in the remainder of this paper that , , and is a nonzero real constant.
3. Coefficient Bounds of the Class
In the following theorem, we determine the initial Taylor coefficients and for the function class
Theorem 3. Let given by (2) belongs to the class Then,
Proof. Let From (15) and (16), we have for some analytic functions such that and for all
It is fairly well known that if and then
Thus, upon comparing the corresponding coefficients in (22) and (23), we have
It follows from (26) and (28) that
If we add (27) and (29), we get
Substituting the value of from (31) in the right-hand side of (32), we deduce that
Moreover, computations using (23), (25), and (33), we find that
Moreover, if we subtract (29) from (27), we obtain
Then, in view of (14) and (31), Eq. (35) becomes
Thus, applying (14), we conclude that
4. Fekete-Szegö Problem for the Function Class
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 , then
This bound is sharp when is real.
In this section, we aim to provide Fekete-Szegö inequalities for functions in the class . These inequalities are given in the following theorem.
Theorem 4. Let given by (2) belongs to the class . Then,
Proof. From (33) and (35) where
Then, in view of (14), we conclude that
which completes the proof of Theorem 4.
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. Let given by (2) belongs to the class . Then,
Corollary 6. Let given by (2) belongs to the class Then,
Corollary 7. Let given by (2) belongs to the class Then,
Corollary 8. Let given by (2) belongs to the class Then,
Corollary 9. Let given by (2) belongs to the class Then,
6. Concluding Remark
By taking and specializing or one can deduce the above results for various subclasses of studied by Bulut et al. [29] and by Altinkaya and Yalcin [32].
Data Availability
Data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors jointly worked on the results and they read and approved the final manuscript.