Two monotonic functions involving gamma function and volume of unit ball

TL;DR

This paper proves the monotonicity of two functions involving the gamma function and the volume of the unit ball in n-dimensional space, contributing to mathematical analysis of special functions.

Contribution

It establishes the monotonicity properties of two gamma-related functions connected to the volume of the unit ball, a novel theoretical result.

Findings

Proved monotonicity of two gamma-related functions

Connected gamma functions to volume of unit ball

Enhanced understanding of special functions in geometry

Abstract

In present paper, we prove the monotonicity of two functions involving the gamma function $\Gamma(x)$ and relating to the $n$-dimensional volume of the unit ball $\mathbb{B}^n$ in $\mathbb{R}^n$.