# §4.18 Inequalities

## Jordan’s Inequality

 4.18.1 $\frac{2x}{\pi}\leq\mathop{\sin\/}\nolimits x\leq x,$ $0\leq x\leq\frac{1}{2}\pi$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\sin\/}\nolimits\NVar{z}$: 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
 4.18.2 $x\leq\mathop{\tan\/}\nolimits x,$ $0\leq x<\frac{1}{2}\pi$, Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\tan\/}\nolimits\NVar{z}$: 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
 4.18.3 $\mathop{\cos\/}\nolimits x\leq\frac{\mathop{\sin\/}\nolimits x}{x}\leq 1,$ $0\leq x\leq\pi$, Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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
 4.18.4 $\pi<\frac{\mathop{\sin\/}\nolimits\!\left(\pi x\right)}{x(1-x)}\leq 4,$ $0. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\sin\/}\nolimits\NVar{z}$: 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

With $z=x+iy$,

 4.18.5 $|\mathop{\sinh\/}\nolimits y|\leq|\mathop{\sin\/}\nolimits z|\leq\mathop{\cosh% \/}\nolimits y,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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
 4.18.6 $|\mathop{\sinh\/}\nolimits y|\leq|\mathop{\cos\/}\nolimits z|\leq\mathop{\cosh% \/}\nolimits y,$
 4.18.7 $|\mathop{\csc\/}\nolimits z|\leq\mathop{\mathrm{csch}\/}\nolimits|y|,$ Symbols: $\mathop{\csc\/}\nolimits\NVar{z}$: cosecant function, $\mathop{\mathrm{csch}\/}\nolimits\NVar{z}$: 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
 4.18.8 $|\mathop{\cos\/}\nolimits z|\leq\mathop{\cosh\/}\nolimits|z|,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\cosh\/}\nolimits\NVar{z}$: 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
 4.18.9 $|\mathop{\sin\/}\nolimits z|\leq\mathop{\sinh\/}\nolimits|z|,$ Symbols: $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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
 4.18.10 $\displaystyle|\mathop{\cos\/}\nolimits z|$ $\displaystyle<2,$ $\displaystyle|\mathop{\sin\/}\nolimits z|$ $\displaystyle\leq\tfrac{6}{5}|z|$, $|z|<1$. Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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

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