单位球体积的单调性和对数凸性
Monotonicity and logarithmic convexity relating to the volume of the unit ball
摘要：
令$Ω_n$代表$n\in\mathbb{N}$的$\mathbb{R}^n$中单位球的体积。在本文中，我们证明了序列$Ω_{n}^{1/(n\ln n)}$是对数凸的，并且序列$\frac{Ω_{n}^{1/(n\ln n)}}{Ω_{n+1}^{1/[(n+1)\ln(n+1)]}}$对于$n\ge2$是严格递减的。此外，还推广和推广了与$Ω_{n}$有关的几个函数的单调性和凹性.
英文摘要：
Let $Ω_n$ 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 $Ω_{n}^{1/(n\ln n)}$ is logarithmically convex and that the sequence $\frac{Ω_{n}^{1/(n\ln n)}}{Ω_{n+1}^{1/[(n+1)\ln(n+1)]}}$ is strictly decreasing for $n\ge2$. In addition, some monotonic and concave properties of several functions relating to $Ω_{n}$ are extended and generalized.
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