Geometric properties of Clausen's Hypergeometric Function $_3F_2(a,b,c;d,e;z)$

TL;DR

This paper introduces an operator involving Clausen's Hypergeometric Function using Hadamard Product and explores its geometric properties based on Taylor coefficients.

Contribution

It presents a new operator defined via Clausen's Hypergeometric Function and analyzes its geometric properties, which is a novel approach in the study of such functions.

Findings

Derived geometric properties of the operator

Connected Taylor coefficients to geometric behavior

Extended understanding of Clausen's Hypergeometric Function applications

Abstract

The Clausen's Hypergeometric Function is given by $${}_3F_2(a,b,c;d,e;z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n(c)_n}{(d)_n(e)_n(1)_n}z^n\, ; \ a,b,c,d,e\in \mathbb{C}$$ provided $d,\, e\, \neq 0,-1,-2,\cdots$ in unit disc $\mathbb{D} =\{z\in \mathbb{C} \,:\, |z|<1\}$. In this paper, an operator $\mathcal{I}_{a,b,c}(f)(z)$ involving Clausen's Hypergeometric Function by means of Hadamard Product is introduced. Geometric properties of $\mathcal{I}_{a,b,c}(f)(z)$ are obtained based on its Taylor's coefficient.