# §4.18 Inequalities

## Jordan’s Inequality

 4.18.1 $\frac{2x}{\pi}\leq\sin x\leq x,$ $0\leq x\leq\frac{1}{2}\pi$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$: sine function and $x$: real variable A&S Ref: 4.3.79 4.3.80 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E1 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.2 $x\leq\tan x,$ $0\leq x<\frac{1}{2}\pi$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\tan\NVar{z}$: tangent function and $x$: real variable A&S Ref: 4.3.80 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E2 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.3 $\cos x\leq\frac{\sin x}{x}\leq 1,$ $0\leq x\leq\pi$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $x$: real variable A&S Ref: 4.3.81 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E3 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.4 $\pi<\frac{\sin\left(\pi x\right)}{x(1-x)}\leq 4,$ $0. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$: sine function and $x$: real variable A&S Ref: 4.3.82 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E4 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4

With $z=x+iy$,

 4.18.5 $|\sinh y|\leq|\sin z|\leq\cosh y,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function, $\sin\NVar{z}$: sine function, $y$: real variable and $z$: complex variable A&S Ref: 4.3.83 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E5 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.6 $|\sinh y|\leq|\cos z|\leq\cosh y,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function, $y$: real variable and $z$: complex variable A&S Ref: 4.3.84 Permalink: http://dlmf.nist.gov/4.18.E6 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.7 $|\csc z|\leq\operatorname{csch}|y|,$ ⓘ Symbols: $\csc\NVar{z}$: cosecant function, $\operatorname{csch}\NVar{z}$: hyperbolic cosecant function, $y$: real variable and $z$: complex variable A&S Ref: 4.3.85 Permalink: http://dlmf.nist.gov/4.18.E7 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.8 $|\cos z|\leq\cosh|z|,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\cosh\NVar{z}$: hyperbolic cosine function and $z$: complex variable A&S Ref: 4.3.86 Permalink: http://dlmf.nist.gov/4.18.E8 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.9 $|\sin z|\leq\sinh|z|,$ ⓘ Symbols: $\sinh\NVar{z}$: hyperbolic sine function, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 4.3.87 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E9 Encodings: TeX, pMML, png See also: Annotations for §4.18, §4.18 and Ch.4
 4.18.10 $\displaystyle|\cos z|$ $\displaystyle<2,$ $\displaystyle|\sin z|$ $\displaystyle\leq\tfrac{6}{5}|z|$, $|z|<1$. ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 4.3.88 Referenced by: §4.18 Permalink: http://dlmf.nist.gov/4.18.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §4.18, §4.18 and Ch.4

For more inequalities see Mitrinović (1964, pp. 101–111), Mitrinović (1970, pp. 235–265), and Bullen (1998, pp. 250–254).