Monotonicity and logarithmic convexity relating to the volume of the unit ball

Details Monotonicity and logarithmic convexity relating to the volume of the unit ball Monotonicity and logarithmic convexity relating to the volume of the unit ball

2009-02-15

arxiv.org/abs/0902.2509v2

Mathematics

Classical Analysis and ODEs

12 pages

Scientific paper

10.1007/s11590-012-0488-2

Let $\Omega_n$ stand for the volume of the unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$. In the present paper, we prove that the sequence $\Omega_{n}^{1/(n\ln n)}$ is logarithmically convex and that the sequence $\frac{\Omega_{n}^{1/(n\ln n)}}{\Omega_{n+1}^{1/[(n+1)\ln(n+1)]}}$ is strictly decreasing for $n\ge2$. In addition, some monotonic and concave properties of several functions relating to $\Omega_{n}$ are extended and generalized.

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