Consolidation of a Certain Discrete Probability Distribution with a Subclass of Bi-Univalent Functions Involving Gegenbauer Polynomials
In this work, we introduce and investigate a new subclass of analytic bi-univalent functions based on subordination conditions between the zero-truncated Poisson distribution and Gegenbauer polynomials. More precisely, we will estimate the first two initial Taylor–Maclaurin coefficients and solve the Fekete–Szegö functional problem for functions belonging to this new subclass.
Let denote the class of all analytic functions defined on the open unit disk and normalized by the conditions and . Thus, each has a Taylor–Maclaurin series expansion of the form
Let the functions and be in the class . We say that the function is subordinate to , written as , if there exists a Schwarz function , which is analytic in withsuch that
Further, let denote the class of all functions that are univalent in . It is worth to mention that if the function is univalent in , then the following equivalence holds (see ):and
It is well known  that every function has an inverse , defined bywhere
A function is said to be bi-univalent in if both and are univalent in . Let denote the class of bi-univalent functions in given by (1). For interesting examples of functions in the class , see [3–10].
In probability theory, the zero-truncated Poisson distribution is a certain discrete probability distribution whose support is the set of positive integers, that is, the zero-truncated Poisson distribution is the same as the Poisson distribution without the zero count . This distribution is also known as the conditional Poisson distribution  or the positive Poisson distribution . The probability density function of the zero-truncated Poisson distribution is given by
Here, let us consider a power series whose coefficients are probabilities of the zero-truncated Poisson distribution, that is,where . By the ratio test, the radius of convergence of this series is infinity.
Define the linear operator bywhere denotes the convolution or Hadamard product of two series (see ).
Orthogonal polynomials have been studied extensively as early as they were discovered by Legendre in 1784 . 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. Recently, many researchers have been exploring bi-univalent functions associated with orthogonal polynomials [16–24]. Very recently, Amourah et al. [25, 26] considered the generating function for Gegenbauer polynomials , which is given bywhere , , and . For fixed , the function is analytic in , so it can be expanded in a Taylor series aswhere is Gegenbauer polynomial of degree (see ).
Obviously, generates nothing when . Therefore, the generating function of the Gegenbauer polynomial is set to befor . Moreover, it is worth to mention that a normalization of to be greater than is desirable [28, 29]. Gegenbauer polynomials can also be defined by the following recurrence relations:with the initial values
Special cases of Gegenbauer polynomials are Chebyshev polynomials when and Legendre polynomials when .
2. The Class
In this section, we introduce a new subclass of involving the new constructed series (10) and Gegenbauer polynomials.
Definition 2.1. A function given by (1) is said to be in the class if the following subordinations are satisfied:where , , , and the function is given by (8).
Upon specializing the parameter , one can get the following new subclass of as illustrated in the following example.
Example 2.1. If , then we have , in which denotes the class of functions given by (1) and satisfying the following conditions.
3. Estimates of the Class
First, we give the coefficient estimates for the class given in Definition 2.1.
Theorem 3.1. Letgiven by (1) belong to the classThen,
Proof. Let . From Definition 2.1, for some analytic functions , such that and , for all , then we can writeFrom the equalities (23) and (24), we obtain thatIt is fairly well known that ifthen (see )Thus, upon comparing the corresponding coefficients in (25) and (26), we haveIt follows from (30) and (32) thatIf we add (31) and (33), we getSubstituting the value of from (35) in the right hand side of (36), we deduce thatMoreover, using (16), (29), and (37), we find thatNow, if we subtract (33) from (31), we obtainThen, in view of (16) and (35), (39) becomesThus, applying (16) and (29), we conclude thatThis completes the proof of Theorem 3.1.
Making use of the values of and , we prove the following Fekete–Szegö inequality for functions in the class .
Theorem 3.2. Letgiven by (1) belong to the class. Then, where
Proof. From (37) and (39),whereThen, in view of (16), we conclude thatwhich completes the proof of Theorem 3.2.
Corresponding essentially to Example 2.1, Theorems 3.1 and 3.2 yield the following consequence.
Corollary 3.3. Letgiven by (1) belong to the class. Then,
In our present investigation, we have introduced a new subclass of normalized analytic and bi-univalent functions associated with the zero-truncated Poisson distribution and Gegenbauer polynomials. For functions belonging to this class, we have derived the estimates of the Taylor–Maclaurin coefficients and and the Fekete–Szegö functional problem. Furthermore, by suitably specializing the parameter , one can deduce the result for the subclass which is defined in Example 2.1.
The results presented in this paper would lead to various other new results for the classes for Chebyshev polynomials and for Legendre polynomials.
The special families examined in this research paper and linked with zero-truncated Poisson distribution and Gegenbauer polynomials could inspire further research related to other aspects, such as families using derivative operator [31, 32] and bi-univalent function families associated with differential operators .
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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