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

Abstract

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

Keywords

• Volume of the unit n-dimensional ball
• Gamma function
• Inequalities
• Logarithmically completely monotonic function