# §4.32 Inequalities

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 $x>0$, ⓘ 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 $\operatorname{arctan}x\leq\tfrac{1}{2}\pi\tanh x,$ $x\geq 0$.

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