§5.6 Inequalities

Keywords:: gamma function, 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 (1997a); for (5.6.4) see Gautschi (1959b) or Laforgia (1984), and for (5.6.5) see Kershaw (1983) and Lorch (2002).
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/5.6.i
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.
Reported 2012-07-30

Throughout this subsection $x>0$ .

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : 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

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

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

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

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

¶ Gautschi’s Inequality

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

¶ Kershaw’s Inequality

Keywords:: inequalities, psi function

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\exp\/}\nolimits z$ : 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

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

§5.6(ii) Complex Variables

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

5.6.6 $|\mathop{\Gamma\/}\nolimits\!\left(x+iy\right)|\leq|\mathop{\Gamma\/}\nolimits\!\left(x\right)|,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $x$ : real variable and $y$ : real variable
Referenced by:: §5.6(ii)
Permalink:: http://dlmf.nist.gov/5.6.E7
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

For $x\geq 0$ ,

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential 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, pMML, png

§5.6 Inequalities

Contents

§5.6(i) Real Variables

¶ Gautschi’s Inequality

¶ Kershaw’s Inequality

§5.6(ii) Complex Variables