Monotonicity and logarithmic convexity relating to the volume of the unit ball

TL;DR

This paper investigates the monotonicity and logarithmic convexity properties of the volume of the unit ball in n-dimensional space, revealing new inequalities and behaviors of related sequences and functions.

Contribution

It proves the logarithmic convexity of the sequence a9_{n}^{1/(n\ln n)} and the decreasing nature of the ratio involving these terms for n, extending known properties of a9_{n}.

Findings

The sequence a9_{n}^{1/(n\ln n)} is logarithmically convex.

The ratio a9_{n}^{1/(n\ln n)} / a9_{n+1}^{1/[(n+1)\ln(n+1)]} is strictly decreasing for n.

Extended and generalized monotonic and concave properties of functions related to a9_{n}.

Abstract

Let $\Omega_n$ stand for the volume of the unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$. In the present paper, we prove that the sequence $\Omega_{n}^{1/(n\ln n)}$ is logarithmically convex and that the sequence $\frac{\Omega_{n}^{1/(n\ln n)}}{\Omega_{n+1}^{1/[(n+1)\ln(n+1)]}}$ is strictly decreasing for $n\ge2$. In addition, some monotonic and concave properties of several functions relating to $\Omega_{n}$ are extended and generalized.