Abstract
Let Ω n denote the volume of the unit ball in \({\mathbb{R}^n}\) for \({n\in\mathbb{N}}\). In the present paper, the authors prove that the sequence \({\Omega_{n}^{1/(n\,{\rm ln}\,n)}}\) is logarithmically convex and that the sequence \({\frac{\Omega_{n}^{1/(n\,{\rm ln}\,n)}}{\Omega_{n+1}^{1/[(n+1)\,{\rm ln}(n+1)]}}}\) is strictly decreasing for n ≥ 2. In addition, some monotonic and concave properties of several functions relating to Ω n are extended and generalized.
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The second author was partially supported by the China Scholarship Council and the Science Foundation of Tianjin Polyteqchnic University.
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Guo, BN., Qi, F. Monotonicity and logarithmic convexity relating to the volume of the unit ball. Optim Lett 7, 1139–1153 (2013). https://doi.org/10.1007/s11590-012-0488-2
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DOI: https://doi.org/10.1007/s11590-012-0488-2