5 Gamma FunctionProperties5.5 Functional Relations5.7 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

Contents

1. §5.6(i) Real Variables
2. §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 $x>0$.

5.6.1
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$\frac{1}{\mathrm{\Gamma }\left(x\right)}+\frac{1}{\mathrm{\Gamma }\left(1/x\right)}\le 2,$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$\frac{1}{{(\mathrm{\Gamma }\left(x\right))}^2}+\frac{1}{{(\mathrm{\Gamma }\left(1/x\right))}^2}\le 2,$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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

Gautschi’s Inequality

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See also:

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

5.6.4
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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

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		$$\exp \left((1-s)\psi \left(x+s^{1/2}\right)\right)\le \frac{\mathrm{\Gamma }\left(x+1\right)}{\mathrm{\Gamma }\left(x+s\right)}\le \exp \left((1-s)\psi \left(x+\frac{1}{2}(s+1)\right)\right),$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $\exp z$: 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).

§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

5.6.6		$$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\le |\mathrm{\Gamma }\left(x\right)|,$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\ge {(\mathrm{sech}\left(\pi y\right))}^{1/2}\mathrm{\Gamma }\left(x\right),$$
$x\ge \frac{1}{2}$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{sech}z$: 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 $b-a\ge 1$, $a\ge 0$, and $z=x+\mathrm{i}y$ with $x>0$,

5.6.8		$$\left|\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\right|\le \frac{1}{{|z|}^{b-a}}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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 $x\ge 0$,

5.6.9		$$|\mathrm{\Gamma }\left(z\right)|\le {(2\pi )}^{1/2}{|z|}^{x-(1/2)}{\mathrm{e}}^{-\pi |y|/2}\exp \left(\frac{1}{6}{|z|}^{-1}\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: 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

5.5 Functional Relations5.7 Series Expansions

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