Applications of q-Hermite Polynomials to Subclasses of Analytic and Bi-Univalent Functions

Abstract

In mathematics, physics, and engineering, orthogonal polynomials and special functions play a vital role in the development of numerical and analytical approaches. This field of study has received a lot of attention in recent decades, and it is gaining traction in current fields, including computational fluid dynamics, computational probability, data assimilation, statistics, numerical analysis, and image and signal processing. In this paper, using q-Hermite polynomials, we define a new subclass of bi-univalent functions. We then obtain a number of important results such as bonds for the initial coefficients of |a2|, |a3|, and |a4|, results related to Fekete–Szegö functional, and the upper bounds of the second Hankel determinant for our defined functions class.

Sharp Bounds of the Fekete–Szegö Problem and Second Hankel Determinant for Certain Bi-Univalent Functions Defined by a Novel q-Differential Operator Associated with q-Limaçon Domain

“…Furthermore, investigated in [29] was the q-analogue of the Salagean operator. For details on the study about this topic, we may refer the readers to [30,31]. Now, using the binomial series…”

“…In [33], the study of analytical and bi-univalent functions is reintroduced; earlier investigations [34][35][36][37][38][39] are among them. New subclasses of bi-univalent function were introduced by a number of authors, and bounds were obtained for the initial coefficients and second Hankel determinant (see [30,31,[40][41][42][43]).…”

Coefficient Inequalities of q-Bi-Univalent Mappings Associated with q-Hyperbolic Tangent Function

“…A widely accepted truth is that a function f (ϑ) ∈ A is classified as a bi-univalent function in E if both f (ϑ) and its inverse function f −1 (w) are separately univalent in E and E t 0 , respectively. The collection of all these bi-univalent functions in E is denoted as Σ and has undergone thorough examination, accompanied by historical context and examples given in [26,27].…”

Certain Subclasses of Univalent Functions Linked with q-Chebyshev Polynomial

“…Since then, the functional has gotten a lot of attention, especially in subclasses of the univalent function family. This problem appears to have piqued the interest of scholars in recent years (see, for example, [26][27][28][29][30][31]). Definition 1.1.…”