Inequalities for the Volume of the Unit Ball in mathbb {R}^{n}, II期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Inequalities for the Volume of the Unit Ball in mathbb {R}^{n}, II

Authors:

Horst Alzer

Affiliation:

1. Morsbacher Str. 10, D-51545, Waldbr?l, Germany

Abstract:

We present several sharp inequalities for the volume of the unit ball in $mathbb {R}^{n}$ ,

$Omega_{n} = frac{{pi ^{n/2}}}{{Gamma(n/2 + 1)}},quad n in {mathbb{N}}$

. One of our theorems states that the double-inequality

$frac{A}{{sqrt n}} leq (n + 1)frac{{Omega_{n + 1}}}{{Omega_n }} - nfrac{{Omega_n}}{{Omega_{n-1} }} < frac{B}{{sqrt n}}$

holds for all n ≥ 2 with the best possible constants

$A = (4 - pi)sqrt{2} = 1.2139 . . .quad{rm and}quad B = frac{1}{2}sqrt{2pi} = 1.2533 . . . .$

This refines and complements a result of Klain and Rota.

Keywords:

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