Bounds for the ratio of two gamma functions--From Wendel's and related inequalities to logarithmically completely monotonic functions

TL;DR

This paper surveys inequalities and monotonicity properties of ratios of gamma and q-gamma functions, analyzing conditions for their complete monotonicity and exploring bounds related to Wendel's inequalities.

Contribution

It provides a comprehensive analysis of inequalities and monotonicity conditions for gamma function ratios, extending previous results and clarifying their properties.

Findings

Identified conditions for logarithmic complete monotonicity of gamma ratio functions

Analyzed inequalities related to Wendel's bounds for gamma functions

Reviewed properties of q-gamma functions and their ratios

Abstract

In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, several complete monotonicity of functions involving ratios of two gamma or $q$-gamma functions, and necessary and sufficient conditions for functions involving ratios of two gamma or $q$-gamma functions to be logarithmically completely monotonic.