§5.6 Inequalities

ⓘ

§5.6(i) Real Variables

ⓘ

Notes:: For (5.6.1) see §5.11(ii); for (5.6.2) see Gautschi (1974); for (5.6.3) see Alzer (1997a); for (5.6.4) see Gautschi (1959b) or Laforgia (1984), and for (5.6.5) see Kershaw (1983) and Lorch (2002).
Referenced by:: Erratum (V1.0.10) for References, Erratum (V1.0.17) for References, Erratum (V1.0.5) for References, Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/5.6.i
Clarification (effective with 1.0.17):: A mention of the references Mortici (2011b, 2013b) was removed because the results contained therein are weaker than (5.6.1).
Addition (effective with 1.0.10):: The reference to Mortici (2013b) has been added at the end of this subsection.
Addition (effective with 1.0.7):: The reference to Mortici (2011b) has been added at the end of this subsection.
Addition (effective with 1.0.5):: References to Alzer (2008), Qi (2008), and Koumandos and Lamprecht (2010) have been added at the end of this subsection. Also, (5.6.5) has been identified as Kershaw’s Inequality.
See also:: Annotations for §5.6 and Ch.5

Throughout this subsection $x>0$ .

5.6.1		$1<(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\Gamma\left(x\right)<e^{1/(12x)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm 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 Ch.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 Ch.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 Ch.5

ⓘ

5.6.4		$x^{1-s}<\frac{\Gamma\left(x+1\right)}{\Gamma\left(x+s\right)}<(x+1)^{1-s},$
$0<s<1$ .
ⓘ 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 Ch.5

ⓘ

5.6.5		$\exp\left((1-s)\psi\left(x+s^{1/2}\right)\right)\leq\frac{\Gamma\left(x+1% \right)}{\Gamma\left(x+s\right)}\leq\exp\left((1-s)\psi\left(x+\tfrac{1}{2}(s+% 1)\right)\right),$
$0<s<1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\exp\NVar{z}$ : exponential function, $s$ : real or complex variable and $x$ : real variable Referenced by: §5.6(i) Permalink: http://dlmf.nist.gov/5.6.E5 Encodings: TeX, pMML, png See also: Annotations for §5.6(i), §5.6(i), §5.6 and Ch.5

For further results see Alzer (2008), Qi (2008), and Koumandos and Lamprecht (2010).

ⓘ

Notes:: For (5.6.6)–(5.6.7) see Carlson (1977b, p. 51); for (5.6.8)–(5.6.9) see Paris and Kaminski (2001, p. 34).
Permalink:: http://dlmf.nist.gov/5.6.ii
See also:: Annotations for §5.6 and Ch.5

5.6.6		$\|\Gamma\left(x+\mathrm{i}y\right)\|\leq\|\Gamma\left(x\right)\|,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{i}$ : imaginary unit, $x$ : 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 Ch.5

5.6.7		$\|\Gamma\left(x+\mathrm{i}y\right)\|\geq(\operatorname{sech}\left(\pi y\right))^% {1/2}\Gamma\left(x\right),$
$x\geq\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\mathrm{i}$ : imaginary unit, $x$ : real variable and $y$ : real variable Referenced by: §5.6(ii) Permalink: http://dlmf.nist.gov/5.6.E7 Encodings: TeX, pMML, png See also: Annotations for §5.6(ii), §5.6 and Ch.5

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 Ch.5

For $x\geq 0$ ,

5.6.9		$\|\Gamma\left(z\right)\|\leq(2\pi)^{1/2}\|z\|^{x-(1/2)}e^{-\pi\|y\|/2}\exp\left(% \tfrac{1}{6}\|z\|^{-1}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $x$ : real variable, $y$ : real variable and $z$ : complex variable Referenced by: §5.6(ii) Permalink: http://dlmf.nist.gov/5.6.E9 Encodings: TeX, pMML, png See also: Annotations for §5.6(ii), §5.6 and Ch.5