Bounds for the ratio of two gamma functions--From Gautschi's and Kershaw's inequalities to completely monotonic functions

TL;DR

This paper surveys and analyzes inequalities and monotonicity properties related to the ratio of two gamma functions, introducing new bounds and exploring complete monotonicity and logarithmic complete monotonicity.

Contribution

It provides a comprehensive review of existing inequalities, investigates monotonicity properties, and presents new bounds for the ratio of two gamma functions and related functions.

Findings

New bounds for the ratio of two gamma functions

Analysis of complete monotonicity of functions involving gamma ratios

Results on logarithmic complete monotonicity and related inequalities

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, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and divided differences of polygamma functions, and related monotonicity results.