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  • Author or Editor: H. Alzer x
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We prove: IfG(n) denotes the geometric mean of the firstn positive integers, then

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$1< 1 + \frac{{G(n)}}{{G(n - 1)}} - \frac{{G(n + 1)}}{{G(n)}}< 1 + \frac{1}{n} - \frac{1}{{n + 1}}< n\frac{{G(n + 1)}}{{G(n)}} - (n - 1)\frac{{G(n)}}{{G(n - 1)}}$$ \end{document}
holds for alln≥3.

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Abstract

We prove: For all natural numbers n and real numbers x ∈ [0, π] we have 548130585k=1n(1)k+1(sin((2k1)x)2k1+sin(2kx)2k).

The sign of equality holds if and only if n = 2 and x = 4π/5.

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