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: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2023-05-01
|
| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-023-02933-1 |
| _version_ | 1797831840082427904 |
|---|---|
| author | Xue-Feng Han Chao-Ping Chen |
| author_facet | Xue-Feng Han Chao-Ping Chen |
| author_sort | Xue-Feng Han |
| collection | DOAJ |
| description | 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 ) $\Omega _{n}^{2}/(\Omega _{n-1}\Omega _{n+1})$ . |
| first_indexed | 2024-04-09T13:59:15Z |
| format | Article |
| id | doaj.art-be802f5c6e13422da5ef2543a493f45f |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-04-09T13:59:15Z |
| publishDate | 2023-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-be802f5c6e13422da5ef2543a493f45f2023-05-07T11:26:58ZengSpringerOpenJournal of Inequalities and Applications1029-242X2023-05-012023111410.1186/s13660-023-02933-1Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$Xue-Feng Han0Chao-Ping Chen1School of Mathematics and Informatics, Henan Polytechnic UniversitySchool of Mathematics and Informatics, Henan Polytechnic UniversityAbstract 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 ) $\Omega _{n}^{2}/(\Omega _{n-1}\Omega _{n+1})$ .https://doi.org/10.1186/s13660-023-02933-1Volume of the unit n-dimensional ballGamma functionInequalitiesLogarithmically completely monotonic function |
| spellingShingle | Xue-Feng Han Chao-Ping Chen Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$ Journal of Inequalities and Applications Volume of the unit n-dimensional ball Gamma function Inequalities Logarithmically completely monotonic function |
| title | Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$ |
| title_full | Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$ |
| title_fullStr | Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$ |
| title_full_unstemmed | Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$ |
| title_short | Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$ |
| title_sort | sharp inequalities related to the volume of the unit ball in r n mathbb r n |
| topic | Volume of the unit n-dimensional ball Gamma function Inequalities Logarithmically completely monotonic function |
| url | https://doi.org/10.1186/s13660-023-02933-1 |
| work_keys_str_mv | AT xuefenghan sharpinequalitiesrelatedtothevolumeoftheunitballinrnmathbbrn AT chaopingchen sharpinequalitiesrelatedtothevolumeoftheunitballinrnmathbbrn |