Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$
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 pr...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2023-05-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-023-02933-1 |
| Summary: | 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 ) $\Omega _{n}^{2}/(\Omega _{n-1}\Omega _{n+1})$ . |
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| ISSN: | 1029-242X |