# §5.6 Inequalities

## §5.6(i) Real Variables

Throughout this subsection $x>0$.

 5.6.1 $1<(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\Gamma\left(x\right) ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $x$: real variable Referenced by: §5.6(i) Permalink: http://dlmf.nist.gov/5.6.E1 Encodings: TeX, pMML, png See also: Annotations for 5.6(i), 5.6 and 5
 5.6.2 $\frac{1}{\Gamma\left(x\right)}+\frac{1}{\Gamma\left(1/x\right)}\leq 2,$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $x$: real variable Referenced by: §5.6(i) Permalink: http://dlmf.nist.gov/5.6.E2 Encodings: TeX, pMML, png See also: Annotations for 5.6(i), 5.6 and 5
 5.6.3 $\frac{1}{(\Gamma\left(x\right))^{2}}+\frac{1}{(\Gamma\left(1/x\right))^{2}}% \leq 2,$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $x$: real variable Referenced by: §5.6(i) Permalink: http://dlmf.nist.gov/5.6.E3 Encodings: TeX, pMML, png See also: Annotations for 5.6(i), 5.6 and 5

### Gautschi’s Inequality

 5.6.4 $x^{1-s}<\frac{\Gamma\left(x+1\right)}{\Gamma\left(x+s\right)}<(x+1)^{1-s},$ $0. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $s$: real or complex variable and $x$: real variable Referenced by: §5.6(i) Permalink: http://dlmf.nist.gov/5.6.E4 Encodings: TeX, pMML, png See also: Annotations for 5.6(i), 5.6(i), 5.6 and 5

### Kershaw’s Inequality

 5.6.5 $\exp\left((1-s)\psi\left(x+s^{1/2}\right)\right)\leq\frac{\Gamma\left(x+1% \right)}{\Gamma\left(x+s\right)}\leq\exp\left((1-s)\psi\left(x+\tfrac{1}{2}(s+% 1)\right)\right),$ $0.

For further results see Alzer (2008), Qi (2008), Koumandos and Lamprecht (2010), and Mortici (2011b, 2013b).

## §5.6(ii) Complex Variables

 5.6.6 $|\Gamma\left(x+\mathrm{i}y\right)|\leq|\Gamma\left(x\right)|,$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $x$: real variable and $y$: real variable A&S Ref: 6.1.26 Referenced by: §5.6(ii) Permalink: http://dlmf.nist.gov/5.6.E6 Encodings: TeX, pMML, png See also: Annotations for 5.6(ii), 5.6 and 5
 5.6.7 $|\Gamma\left(x+\mathrm{i}y\right)|\geq(\operatorname{sech}\left(\pi y\right))^% {1/2}\Gamma\left(x\right),$ $x\geq\tfrac{1}{2}$.

For $b-a\geq 1$, $a\geq 0$, and $z=x+iy$ with $x>0$,

 5.6.8 $\left|\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+b\right)}\right|\leq\frac{1}% {|z|^{b-a}}.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable Referenced by: §5.6(ii) Permalink: http://dlmf.nist.gov/5.6.E8 Encodings: TeX, pMML, png See also: Annotations for 5.6(ii), 5.6 and 5

For $x\geq 0$,

 5.6.9 $|\Gamma\left(z\right)|\leq(2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp\left(% \tfrac{1}{6}|z|^{-1}\right).$