Complete monotonicity of a function involving the gamma function and applications

TL;DR

This paper establishes conditions for a function involving the gamma function to be completely monotonic and applies these to derive bounds for gamma, harmonic numbers, and factorials.

Contribution

It provides necessary and sufficient conditions for complete monotonicity of gamma-related functions and applies these to obtain bounds on gamma, harmonic numbers, and factorials.

Findings

Derived conditions for complete monotonicity involving the gamma function

Established bounds for gamma function, harmonic numbers, and factorials

Applied theoretical results to practical inequalities

Abstract

In the article we present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function $\Gamma(x)$, the $n$-th harmonic number $\sum_{k=1}^n\frac1k$, and the factorial $n!$.