Close-to-Convexity properties of Clausen's Hypergeometric Function $_3F_2(a,b,c;d,e;z)$

TL;DR

This paper establishes conditions on parameters of Clausen's hypergeometric function $_3F_2$ to ensure it belongs to the class of close-to-convex functions, contributing to geometric function theory.

Contribution

It provides new parameter conditions for Clausen's hypergeometric function to be close-to-convex, expanding understanding of its geometric properties.

Findings

Identifies parameter ranges for close-to-convexity

Links hypergeometric functions to geometric function classes

Enhances theoretical understanding of $_3F_2$ functions

Abstract

In this paper, we obtain various conditions on the parameters $a,\, b,\, c\,, d$ and $e$ for which the hypergeometric functions $z\,_3F_2(a,b,c;d,e;z)$ to be in the class of all close-to-convex function with respect to some well known convex functions.