DLMF: GA.6 Inequalities

§GA.6. Inequalities

Sources:

Gautschi(1974), Alzer(1997), Laforgia(1984), Kershaw(1983), Lorch(2002), Carlson(1977)(§3.10), Paris and Kaminski(2001)(§2.1.3). For (GA.6.1) see §GA.11(ii).

Keywords:

gamma function, inequalities

§GA.6(i)Real Variables
§GA.6(ii)Complex Variables

§GA.6(i). Real Variables

Notes:

Throughout this subsection $x>0$ .

GA.6.1 $1<(2\pi)^{{-1/2}}x^{{(1/2)-x}}e^{x}\Gamma\!\left(x\right)<e^{{1/(12x)}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $x$ : real variable
Referenced by:: §GA.6(i), §GA.6
Encodings:: pMathML, png, TeX

GA.6.2 $\frac{1}{\Gamma\!\left(x\right)}+\frac{1}{\Gamma\!\left(1/x\right)}\le 2,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $x$ : real variable
Referenced by:: §GA.6(i)
Encodings:: pMathML, png, TeX

GA.6.3 $\frac{1}{(\Gamma\!\left(x\right))^{2}}+\frac{1}{(\Gamma\!\left(1/x\right))^{2}}\le 2,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $x$ : real variable
Referenced by:: §GA.6(i)
Encodings:: pMathML, png, TeX

GA.6.4 $x^{{1-s}}<\frac{\Gamma\!\left(x+1\right)}{\Gamma\!\left(x+s\right)}<(x+1)^{{1-s}},$ $0<s<1$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $s$ : real or complex variable and $x$ : real variable
Referenced by:: §GA.6(i)
Encodings:: pMathML, png, TeX

GA.6.5 $\mathrm{exp}\!\left((1-s)\psi\!\left(x+s^{{1/2}}\right)\right)\le\frac{\Gamma\!\left(x+1\right)}{\Gamma\!\left(x+s\right)}\le\mathrm{exp}\!\left((1-s)\psi\!\left(x+\tfrac{1}{2}(s+1)\right)\right),$ $0<s<1$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $\psi\!\left(z\right)$ : Psi or digamma function, $s$ : real or complex variable and $x$ : real variable
Referenced by:: §GA.6(i)
Encodings:: pMathML, png, TeX

§GA.6(ii). Complex Variables

Notes:

Carlson(1977)(p. 51)

Paris and Kaminski(2001)(p. 34)

GA.6.6 $|\Gamma\!\left(x+iy\right)|\le|\Gamma\!\left(x\right)|,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $x$ : real variable and $y$ : real variable
A&S Ref:: 6.1.26
Referenced by:: §GA.6(ii)
Encodings:: pMathML, png, TeX

GA.6.7 $|\Gamma\!\left(x+iy\right)|\ge(\mathrm{sech}\!\left(\pi y\right))^{{1/2}}\Gamma\!\left(x\right),$ $x\ge\tfrac{1}{2}$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $x$ : real variable and $y$ : real variable
Referenced by:: §GA.6(ii)
Encodings:: pMathML, png, TeX

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

GA.6.8 $\left|\frac{\Gamma\!\left(z+a\right)}{\Gamma\!\left(z+b\right)}\right|\le\frac{1}{|z|^{{b-a}}}.$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.6(ii)
Encodings:: pMathML, png, TeX

For $x\ge 0$ ,

GA.6.9 $|\Gamma\!\left(z\right)|\le(2\pi)^{{1/2}}|z|^{{x-(1/2)}}e^{{-\pi|y|/2}}\mathrm{exp}\!\left(\tfrac{1}{6}|z|^{{-1}}\right).$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $x$ : real variable, $y$ : real variable and $z$ : complex variable
Referenced by:: §GA.6(ii)
Encodings:: pMathML, png, TeX

§GA.6. Inequalities

Contents

§GA.6(i). Real Variables

§GA.6(ii). Complex Variables