Murugusundaramoorthy Gangadharan

Bi-Bazilevic functions based on Hurwitz-Lerch Zeta function associated with exponential Pareto distribution

Introduction

Bi-bazilevic functions based on hurwitz-lerch zeta function associated with exponential pareto distribution. Explore new bi-univalent Bi-Bazilevic functions based on Hurwitz-Lerch Zeta & exponential Pareto distribution. Provides Taylor-Maclaurin coefficient estimates & Fekete-Szegö inequality results.

Abstract

In this paper, we introduce and investigate new subclass of bi-univalent functions defined in the open unit disk, which are based on Hurwitz-Lerch Zeta function associated with exponential Pareto distribution , satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclass. Several new consequences of the results are also pointed out.Additionally we discussed Fekete-Szegö inequality results

Review

This paper presents a novel exploration within the field of geometric function theory, specifically focusing on the class of bi-univalent functions. The authors introduce and investigate a new subclass of these functions, termed "Bi-Bazilevic functions," which are uniquely constructed by integrating the Hurwitz-Lerch Zeta function associated with the exponential Pareto distribution. This interdisciplinary approach, combining concepts from special functions and probability distributions within the framework of complex analysis, offers a fresh perspective on defining and analyzing properties of univalent and bi-univalent functions, addressing a current area of active research. The methodology employed involves defining this new subclass within the open unit disk, utilizing subordinate conditions to establish its properties. A primary objective of the work is to derive estimates for the Taylor-Maclaurin coefficients, specifically $|a_2|$ and $|a_3|$, which are fundamental in characterizing the geometric behavior of functions in this class. Furthermore, the paper delves into the Fekete-Szegö inequality, a significant result in geometric function theory that provides further insights into the coefficient bounds. The abstract also notes that several new consequences of these results are discussed, implying a thorough analysis of the derived inequalities and their implications for the newly defined subclass. Overall, this research appears to make a valuable contribution to the ongoing study of bi-univalent functions by introducing a distinct method for their generation and analysis. The innovative use of the Hurwitz-Lerch Zeta function in conjunction with the exponential Pareto distribution creates a fertile ground for exploring new subclasses and their inherent properties. The findings regarding coefficient estimates and the Fekete-Szegö inequality will be of interest to researchers working in geometric function theory, offering foundational results for this specific subclass and potentially paving the way for future investigations into other special functions or probability distributions in similar contexts.

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