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

Abstract

Let $Ω_n$ 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 $Ω_{n}^{1/(n\ln n)}$ is logarithmically convex and that the sequence $\frac{Ω_{n}^{1/(n\ln n)}}{Ω_{n+1}^{1/[(n+1)\ln(n+1)]}}$ is strictly decreasing for $n\ge2$. In addition, some monotonic and concave properties of several functions relating to $Ω_{n}$ are extended and generalized.

12 pages