A ratio of many gamma functions and its properties with applications

TL;DR

This paper introduces a new ratio of multiple gamma functions, explores its mathematical properties such as monotonicity and convexity, and applies these findings to derive inequalities involving multinomial coefficients and multivariate beta functions.

Contribution

The paper establishes a novel ratio of many gamma functions and analyzes its properties, leading to new inequalities in special functions.

Findings

The ratio exhibits monotonicity and convexity properties.

Derived inequalities for multinomial coefficients.

Established properties of the ratio as a Bernstein function.

Abstract

In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.