Complete monotonicity of a function involving a ratio of gamma functions and applications

TL;DR

This paper establishes conditions for a gamma function ratio to be logarithmically completely monotonic and applies these results to derive and extend inequalities related to the volume of unit balls in n-dimensional space.

Contribution

It provides necessary and sufficient conditions for the complete monotonicity of a gamma ratio, extending previous results and applying them to geometric inequalities.

Findings

Derived new inequalities for the volume of the unit ball in .

Generalized existing monotonicity results for gamma function ratios.

Extended the scope of inequalities involving gamma functions.

Abstract

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.