Mapping properties of the zero-balanced hypergeometric functions

TL;DR

This paper investigates the convexity properties of zero-balanced hypergeometric functions, establishing conditions for convexity and describing their image domains, thus advancing understanding of their geometric behavior.

Contribution

It provides new results on the convexity and image domains of zero-balanced hypergeometric functions under specific parameter conditions.

Findings

Order of convexity of hypergeometric functions determined

Image domains are convex and bounded by horizontal lines

Addresses a problem posed by Ponnusamy and Vuorinen

Abstract

In the present paper, the order of convexity of z\Gauss(a,b;c;z) is first given under some conditions on the positive real parameters a, b and c. Then we show that the image domains of the unit disc \D under some shifted zero-balanced hypergeometric functions z\Gauss(a,b;a+b;z) are convex and bounded by two horizontal lines which solves the problem raised by Ponnusamy and Vuorinen in \cite{PonVuor:2001