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Inequalities for 1/(1-cos(x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(1-\cos(x) )$$\end{document} and its derivatives
被引：0
作者：
H. Alzer
[1
]
H. L. Pedersen
[2
]
机构：
[1] University of Copenhagen,Department of Mathematical Sciences
关键词：
completely and absolutely monotonic;
trigonometric function;
inequality;
sub- and superadditive;
Bernoulli number;
Euler number;
Fejér’s sine polynomial;
partial
fraction decomposition;
26A48;
26D05;
11B68;
D O I：
10.1007/s10476-025-00069-6
中图分类号：
学科分类号：
摘要：
We prove that the function g(x)=1/(1-cos(x))\documentclass[12pt]{minimal}
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\begin{document}$$g(x)= 1 / ( 1 - \cos(x) )$$\end{document} is completely monotonic on (0,π]\documentclass[12pt]{minimal}
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\begin{document}$$(0,\pi]$$\end{document} and absolutely monotonic on [π,2π)\documentclass[12pt]{minimal}
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\begin{document}$$[\pi, 2\pi)$$\end{document}, and we determine the best possible bounds λn\documentclass[12pt]{minimal}
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\begin{document}$$\lambda_n$$\end{document} and μn\documentclass[12pt]{minimal}
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\begin{document}$$\mu_n$$\end{document} such that the inequalities
λn≤g(n)(x)+g(n)(y)-g(n)(x+y)(n≥0even)\documentclass[12pt]{minimal}
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\begin{document}$$
\lambda_n \leq g^{(n)}(x)+g^{(n)}(y)-g^{(n)}(x+y) \quad (n \geq 0 \ \mbox{even})
$$\end{document}
and
μn≤g(n)(x+y)-g(n)(x)-g(n)(y)(n≥1odd)\documentclass[12pt]{minimal}
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\begin{document}$$
\mu_n \leq g^{(n)}(x+y)-g^{(n)}(x)-g^{(n)}(y) \quad (n \geq 1 \ \mbox{odd})
$$\end{document}
hold for all x,y∈(0,π)\documentclass[12pt]{minimal}
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\begin{document}$$x,y\in (0,\pi)$$\end{document} with x+y≤π\documentclass[12pt]{minimal}
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\begin{document}$$x+y\leq \pi$$\end{document}.
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页码：63 / 73
页数：10
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