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

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 $n\ge 2$. In addition, some monotonic and concave properties of several functions relating to ${\Omega }_n$ are extended and generalized.

12 pages

References in corpus (4)

• Complete monotonicity of some functions involving polygamma functions
• Two new proofs of the complete monotonicity of a function involving the psi function
• Some logarithmically completely monotonic functions related to the gamma function
• An extension of an inequality for ratios of gamma functions

Cited by in corpus (8)

• Bounds for the ratio of two gamma functions--From Wendel's limit to Elezović-Giordano-Pečarić's theorem
• A logarithmically completely monotonic function involving the ratio of gamma functions
• An extension of an inequality for ratios of gamma functions
• A completely monotonic function involving the gamma and tri-gamma functions
• Topics in special functions III
• A Pick function related to the sequence of volumes of the unit ball in n-space
• Reflections on Ramanujan's Mathematical Gems
• A completely monotonic function used in an inequality of Alzer