§ 5.6. Inequalities

Permalink:: http://dlmf.nist.gov/5.6

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

§ 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 (1997); for (5.6.4) see Gautschi (1959) or Laforgia (1984), and for (5.6.5) see Kershaw (1983) and Lorch (2002).
Permalink:: http://dlmf.nist.gov/5.6.SS1

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:: $\Gamma\!\left(z\right)$ : Gamma function and $x$ : real variable
Referenced by:: §5.6(i)
Permalink:: http://dlmf.nist.gov/5.6.E1
Encodings:: TeX, pMathML, png

5.6.2 $\frac{1}{\Gamma\!\left(x\right)}+\frac{1}{\Gamma\!\left(1/x\right)}\le 2,$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $x$ : real variable
Referenced by:: §5.6(i)
Permalink:: http://dlmf.nist.gov/5.6.E2
Encodings:: TeX, pMathML, png

5.6.3 $\frac{1}{(\Gamma\!\left(x\right))^{2}}+\frac{1}{(\Gamma\!\left(1/x\right))^{2}}\le 2,$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $x$ : real variable
Referenced by:: §5.6(i)
Permalink:: http://dlmf.nist.gov/5.6.E3
Encodings:: TeX, pMathML, png

¶ Gautschi's Inequality

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

Symbols:: $\Gamma\!\left(z\right)$ : 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, pMathML, png

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\psi\!\left(z\right)$ : Psi or digamma 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, pMathML, png

§ 5.6(ii). Complex Variables

Notes:: For (5.6.6)–(5.6.7) see Carlson (1977, p. 51); for (5.6.8)–(5.6.9) see Paris and Kaminski (2001, p. 34).
Permalink:: http://dlmf.nist.gov/5.6.SS2

5.6.6 $|\Gamma\!\left(x+iy\right)|\le|\Gamma\!\left(x\right)|,$

Symbols:: $\Gamma\!\left(z\right)$ : 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, pMathML, png

5.6.7 $|\Gamma\!\left(x+iy\right)|\ge(\mathrm{sech}\!\left(\pi y\right))^{{1/2}}\Gamma\!\left(x\right),$ $x\ge\tfrac{1}{2}$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $x$ : real variable and $y$ : real variable
Referenced by:: §5.6(ii)
Permalink:: http://dlmf.nist.gov/5.6.E7
Encodings:: TeX, pMathML, png

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

5.6.8 $\left|\frac{\Gamma\!\left(z+a\right)}{\Gamma\!\left(z+b\right)}\right|\le\frac{1}{|z|^{{b-a}}}.$

Symbols:: $\Gamma\!\left(z\right)$ : 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, pMathML, png

For $x\ge 0$ ,

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $x$ : real variable, $y$ : real variable and $z$ : complex variable
Referenced by:: §5.6(ii)
Permalink:: http://dlmf.nist.gov/5.6.E9
Encodings:: TeX, pMathML, png

§ 5.6. Inequalities

Contents

§ 5.6(i). Real Variables

¶ Gautschi's Inequality

§ 5.6(ii). Complex Variables