Bounds for the ratio of two gamma functions--From Wendel's limit to Elezovi\'c-Giordano-Pe\v{c}ari\'c's theorem

TL;DR

This paper surveys classical and recent bounds on the ratio of two gamma functions, analyzing known inequalities and introducing new advances in the field.

Contribution

It provides a comprehensive review of existing results and presents recent developments in bounding the ratio of two gamma functions.

Findings

Analysis of classical inequalities like Wendel's and Kershaw's

Introduction of recent bounds and monotonicity properties

Comparison of various bounds and their applications

Abstract

In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel's, Gurland's, Kazarinoff's, Gautschi's, Watson's, Chu's, Lazarevi\'c-Lupa\c{s}'s, Kershaw's and Elezovi\'c-Giordano-Pe\v{c}ari\'c's inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years.