Sharp inequalities related to the volume of the unit ball in mathbb{R}^{n}

Abstract

Let Omega _{n}=pi ^{n/2}/Gamma (frac{n}{2}+1) (n in mathbb{N}) denote the volume of the unit ball in mathbb{R}^{n}. In this paper, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions is presented, which yields a sharp double inequality for the quantity Omega _{n}^{2}/(Omega _{n-1}Omega _{n+1}). Also, we establish new sharp inequalities for the quantity Omega _{n}^{2}/(Omega _{n-1}Omega _{n+1}).