Universal convexity and range problems of shifted hypergeometric functions

TL;DR

This paper investigates the range and convexity properties of shifted hypergeometric functions with real parameters, providing solutions to longstanding problems and establishing conditions for their universal convexity.

Contribution

It solves the range problems for shifted hypergeometric functions and establishes new conditions for their universal convexity, extending previous work on starlikeness.

Findings

Solved the range problems for functions $f$ and $g$.

Established conditions for the universal convexity of shifted hypergeometric functions.

Extended the theory of universal convexity beyond the case $b=1$.

Abstract

In the present paper, we study the shifted hypergeometric function $f(z)=z\Gauss(a,b;c;z)$ for real parameters with $0<a\le b\le c$ and its variant $g(z)=z\Gauss(a,b;c;z^2).$ Our first purpose is to solve the range problems for $f$ and $g$ posed by Ponnusamy and Vuorinen in their 2001 paper. Ruscheweyh, Salinas and Sugawa developed in their 2009 paper the theory of universal prestarlike functions on the slit domain $\C\setminus[1,+\infty)$ and showed universal starlikeness of $f$ under some assumptions on the parameters. However, there has been no systematic study of universal convexity of the shifted hypergeometric functions except for the case $b=1.$ Our second purpose is to show universal convexity of $f$ under certain conditions on the parameters.