More supplements to a class of logarithmically completely monotonic functions associated with the gamma function

TL;DR

This paper establishes conditions for certain gamma-related functions to be logarithmically completely monotonic, leading to new inequalities for psi, polygamma, and gamma function ratios, advancing understanding of gamma function properties.

Contribution

It provides necessary and sufficient conditions for gamma-related functions to be logarithmically completely monotonic, which was not previously known.

Findings

Derived inequalities for psi and polygamma functions

Bounded ratios of gamma functions

Characterized monotonicity conditions for gamma-related functions

Abstract

In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on $(0,\infty)$. As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived.