Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions

Ala Amourah,Basem Aref Frasin,Feras Yousef,Morad Ahmad

doi:10.3390/sym14010147

Ala Amourah, Basem Aref Frasin + Show 2 more

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https://doi.org/10.3390/sym14010147

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Abstract

In the present analysis, we aim to construct a new subclass of analytic bi-univalent functions defined on symmetric domain by means of the Pascal distribution series and Gegenbauer polynomials. Thereafter, we provide estimates of Taylor–Maclaurin coefficients a2 and a3 for functions in the aforementioned class, and next, we solve the Fekete–Szegö functional problem. Moreover, some interesting findings for new subclasses of analytic bi-univalent functions will emerge by reducing the parameters in our main results.

Highlights

Introduction and PreliminariesIn statistics and probability, distributions of random variables play a basic role and are used extensively to describe and model a lot of real life phenomena; they describe the distribution of the probabilities over the random variable values [1]
If we have two possible outcomes or in our random experiment and we are interested in how many independent times we need to repeat this random experiment until we achieve the first success, the random variable X which represents this number of trials has a geometric distribution
Motivated essentially by the work of Amourah et al [13], we construct, a new subclass of bi-univalent functions governed by the Pascal distribution series and Gegenbauer polynomials

Summary

Introduction and Preliminaries

Distributions of random variables play a basic role and are used extensively to describe and model a lot of real life phenomena; they describe the distribution of the probabilities over the random variable values [1]. The probability density function of a discrete random variable X that follows a Pascal distribution reads as x−1. Let Σ denote the class of all functions f ∈ A that are bi-univalent in U given by (7). Motivated essentially by the work of Amourah et al [13], we construct, a new subclass of bi-univalent functions governed by the Pascal distribution series and Gegenbauer polynomials. A function f is said to be in the class GΣ ( x, α, γ) = GΣ ( x, α, 0, γ), if the following subordinations are fulfilled:.

Main Results

Corollaries and Consequences

Concluding Remarks