Some logarithmically completely monotonic functions related to the gamma function

TL;DR

This paper investigates the logarithmic complete monotonicity of various functions involving the gamma function, extending and generalizing known results, and establishing fundamental properties of such functions.

Contribution

It introduces new results on the monotonicity properties of gamma-related functions, broadening the understanding of their mathematical behavior and relationships.

Findings

Identified conditions for logarithmic complete monotonicity of gamma-based functions

Extended known monotonicity results to more general functions

Established basic properties of logarithmically completely monotonic functions

Abstract

In this article, logarithmically complete monotonicity properties of some functions such as $\frac1{[\Gamma(x+1)]^{1/x}}$, $\frac{[{\Gamma(x+\alpha+1)}]^{1/(x+\alpha)}}{[{\Gamma(x+1)}]^{1/x}}$, $\frac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}$ and $\frac{[\Gamma(x+1)]^{1/x}}{x^\alpha}$ defined in $(-1,\infty)$ or $(0,\infty)$ for given real number $\alpha\in\tr$ are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.