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Inequalities and Asymptotic Expansions Related to the Volume of the Unit Ball in \(\pmb {\mathbb {R}}^{{{\varvec{n}}}}\)

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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, we present asymptotic expansions and inequalities related to \(\Omega _{n}\) and the quantities:

$$\begin{aligned} \frac{\Omega _{n-1}}{\Omega _{n}}, \quad \frac{\Omega _{n}}{\Omega _{n-1}+\Omega _{n+1}} \quad \text {and}\quad \frac{\Omega _n^{1/n}}{\Omega _{n+1}^{1/(n+1)}}. \end{aligned}$$

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Authors and Affiliations

  1. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454000, Henan Province, China

    Chao-Ping Chen

  2. Division of Computing and Mathematics, Abertay University, Dundee, DD1 1HG, UK

    Richard B. Paris

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  1. Chao-Ping Chen
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  2. Richard B. Paris
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Correspondence to Chao-Ping Chen.

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Chen, CP., Paris, R.B. Inequalities and Asymptotic Expansions Related to the Volume of the Unit Ball in \(\pmb {\mathbb {R}}^{{{\varvec{n}}}}\). Results Math 74, 44 (2019). https://doi.org/10.1007/s00025-019-0967-1

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