Inequalities for Zero-Balanced Gaussian hypergeometric function

TL;DR

This paper investigates inequalities involving zero-balanced Gaussian hypergeometric functions, establishing monotonicity properties and identifying the largest parameter range for which certain inequalities hold across the interval (0,1).

Contribution

It introduces new inequalities and monotonicity results for specific Gaussian hypergeometric functions, expanding understanding of their comparative behavior.

Findings

Identified the largest delta for which the inequality holds.

Established monotonicity properties of hypergeometric function combinations.

Provided bounds and comparison results for these functions.

Abstract

In this paper, we consider the monotonicity of certain combinations of the Gaussian hypergeometric functions $F(a-1,b;a+b;1-x^c)$ and $F(a-1-\delta,b+\delta;a+b;1-x^d)$ on $(0,1)$ for $\delta\in(a-1,0)$, and study the problem of comparing these two functions, thus get the largest value $\delta_1=\delta_1(a,c,d)$ such that the inequality $F(a-1,b;a+b;1-x^c)<F(a-1-\delta,b+\delta;a+b;1-x^d)$ holds for all $x\in (0,1)$.