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DLMF: §4.32 Inequalities ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions

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4 Elementary Functions Hyperbolic Functions 4.31 Special Values and Limits 4.33 Maclaurin Series and Laurent Series

§4.32 Inequalities

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Keywords:: hyperbolic functions, inequalities
Permalink:: http://dlmf.nist.gov/4.32
See also:: Annotations for Ch.4

For $x$ real,

4.32.1		$\cosh x\leq\left(\frac{\sinh x}{x}\right)^{3},$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $x$ : real variable Permalink: http://dlmf.nist.gov/4.32.E1 Encodings: TeX, pMML, png See also: Annotations for §4.32 and Ch.4

4.32.2		$\sin x\cos x<\tanh x<x,$
$x>0$ ,
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\sin\NVar{z}$ : sine function and $x$ : real variable Permalink: http://dlmf.nist.gov/4.32.E2 Encodings: TeX, pMML, png See also: Annotations for §4.32 and Ch.4

4.32.3		$\|\cosh x-\cosh y\|\geq\|x-y\|\sqrt{\sinh x\sinh y},$
$x>0$ , $y>0$ ,
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/4.32.E3 Encodings: TeX, pMML, png See also: Annotations for §4.32 and Ch.4

4.32.4		$\operatorname{arctan}x\leq\tfrac{1}{2}\pi\tanh x,$
$x\geq 0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\tanh\NVar{z}$ : hyperbolic tangent function, $\operatorname{arctan}\NVar{z}$ : arctangent function and $x$ : real variable Permalink: http://dlmf.nist.gov/4.32.E4 Encodings: TeX, pMML, png See also: Annotations for §4.32 and Ch.4

For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).

4.31 Special Values and Limits 4.33 Maclaurin Series and Laurent Series

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