5 Gamma Function Properties 5.5 Functional Relations 5.7 Series Expansions

§5.6 Inequalities

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Keywords:: gamma function, inequalities, maxima and minima
Permalink:: http://dlmf.nist.gov/5.6
See also:: Annotations for Ch.5

§5.6(i) Real Variables
§5.6(ii) Complex Variables

§5.6(i) Real Variables

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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 .

5.6.1 $1<(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\Gamma\left(x\right)<e^{1/(12x)},$

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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 : 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 $\frac{1}{\Gamma\left(x\right)}+\frac{1}{\Gamma\left(1/x\right)}\leq 2,$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function and : 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 $\frac{1}{(\Gamma\left(x\right))^{2}}+\frac{1}{(\Gamma\left(1/x\right))^{2}}% \leq 2,$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function and : 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

Gautschi’s Inequality

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See also:: Annotations for §5.6(i), §5.6 and Ch.5

5.6.4 $x^{1-s}<\frac{\Gamma\left(x+1\right)}{\Gamma\left(x+s\right)}<(x+1)^{1-s},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, : real or complex variable and : 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

Kershaw’s Inequality

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Keywords:: inequalities, psi function
See also:: Annotations for §5.6(i), §5.6 and Ch.5

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\exp\NVar{z}$ : exponential function, : real or complex variable and : 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).

§5.6(ii) Complex Variables

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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