# Complete Monotonicity and Inequalities of Functions Involving $\Gamma$-function

## TL;DR

This paper studies the complete monotonicity of functions involving the gamma function and derives inequalities related to gamma and beta functions, which are relevant in probability theory.

## Contribution

It introduces new results on the complete monotonicity of gamma-related functions and establishes related inequalities, expanding the theoretical understanding of these functions.

## Key findings

- Established conditions for complete monotonicity of gamma functions
- Derived new inequalities involving gamma and beta functions
- Applied inequalities to problems in probability theory

## Abstract

In this paper, we investigate the complete monotonicity of some functions involving gamma function. Using the monotonic properties of these functions, we derived some inequalities involving gamma and beta functions. Such inequalities arising in probability theory.