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

Abstract

AbstractLet$\Omega _{n}=\pi ^{n/2}/\Gamma (\frac{n}{2}+1)$Ωn=πn/2/Γ(n2+1)($n \in \mathbb{N}$n∈N) denote the volume of the unit ball in$\mathbb{R}^{n}$Rn. 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$\Omega _{n}^{2}/(\Omega _{n-1}\Omega _{n+1})$Ωn2/(Ωn−1Ωn+1). Also, we establish new sharp inequalities for the quantity$\Omega _{n}^{2}/(\Omega _{n-1}\Omega _{n+1})$Ωn2/(Ωn−1Ωn+1).