An extension of an inequality for ratios of gamma functions

TL;DR

This paper proves a new inequality involving ratios of gamma functions, extending previous results and resolving an open problem by establishing the inequality's validity and optimality conditions.

Contribution

It extends an existing inequality for gamma function ratios, providing the best possible power and conditions for validity and reversing.

Findings

The inequality holds for x > 1 and is reversed for x < 1.

The power 1/2 is proven to be optimal.

The result generalizes and improves previous bounds.

Abstract

In this paper, we prove that for $x+y>0$ and $y+1>0$ the inequality {equation*} \frac{[\Gamma(x+y+1)/\Gamma(y+1)]^{1/x}}{[\Gamma(x+y+2)/\Gamma(y+1)]^{1/(x+1)}} <\biggl(\frac{x+y}{x+y+1}\biggr)^{1/2} {equation*} is valid if $x>1$ and reversed if $x<1$ and that the power $\frac12$ is the best possible, where $\Gamma(x)$ is the Euler gamma function. This extends the result in [Y. Yu, \textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl. \textbf{352} (2009), no.~2, 967\nobreakdash--970.] and resolves an open problem posed in [B.-N. Guo and F. Qi, \emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.].