On quotients and differences of hypergeometric functions

TL;DR

This paper derives optimal bounds for quotients and differences of Gaussian hypergeometric functions, enhancing understanding of their behavior under various parameter conditions.

Contribution

It provides the best possible bounds for these hypergeometric function expressions based on different parameter assumptions.

Findings

Established sharp bounds for quotients of hypergeometric functions.

Derived optimal bounds for differences of hypergeometric functions.

Analyzed bounds under various parameter constraints.

Abstract

For Gaussian hypergeometric functions $F(x)= F(a,b;c;x),$ $a,b,c>0,$ we consider the quotient $ Q_F(x,y)= (F(x)+F(y))/F(z)$ and the difference $ D_F(x,y)= F(x)+F(y)-F(z)$ for $0<x,y<1$ with $z=x+y-xy \,.$ We give best possible bounds for both expressions under various hypotheses about the parameter triple $(a,b;c)\,.$