Univalent Functions with Non-Negative Coefficients involving Clausen's Hypergeometric Function

TL;DR

This paper establishes parameter conditions for a specific hypergeometric function to belong to certain geometric function classes and explores its image properties using convolution operators in the unit disk.

Contribution

It provides the first comprehensive criteria for $_3F_2$ hypergeometric functions to be in classes $ ext{M}^*( ext{lambda}, ext{alpha})$ and $ ext{N}^*( ext{lambda}, ext{alpha})$, and analyzes their image sets.

Findings

Derived necessary and sufficient conditions for hypergeometric functions to belong to specific classes.

Characterized the image of $_3F_2$ functions in the class $ ext{R}^ au(A,B)$.

Applied convolution operators to analyze function properties in the unit disk.

Abstract

In this work, we derived the necessary and sufficient conditions on parameters for $_3F_2(^{a,b,c}_{b+1,c+1};z)$ Hypergeometric Function to be in the classes $\mathcal{M}^{\ast}(\lambda,\alpha)$ and $\mathcal{N}^{\ast}(\lambda,\alpha)$ and information regarding the image of function $_3F_2(^{a,b,c}_{b+1,c+1};z)$ belonging to $\mathcal{R}^{\tau}(A,B)$ by applying the convolution operator in open unit disc $\mathbb{D} =\{z:\, |z|<1\}$.