8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.9 Continued Fractions 8.11 Asymptotic Approximations and Expansions

§8.10 Inequalities

ⓘ

Keywords:: incomplete gamma functions, inequalities
Notes:: See Olver (1997b, pp. 66–67) and Luke (1969b, pp. 195, 201). For (8.10.10)–(8.10.13), see Gautschi (1959b), Alzer (1997b), and Vietoris (1983).
Referenced by:: Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/8.10
Addition (effective with 1.0.10):: A reference to Neuman (2013) was added after (8.10.4)
See also:: Annotations for Ch.8

8.10.1		$x^{1-a}e^{x}\Gamma\left(a,x\right)\leq 1,$
$x>0$ , $0<a\leq 1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter Referenced by: §8.10 Permalink: http://dlmf.nist.gov/8.10.E1 Encodings: TeX, pMML, png See also: Annotations for §8.10 and Ch.8

8.10.2		$\gamma\left(a,x\right)\geq\frac{x^{a-1}}{a}(1-e^{-x}),$
$x>0$ , $0<a\leq 1$ .
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter Referenced by: §8.10 Permalink: http://dlmf.nist.gov/8.10.E2 Encodings: TeX, pMML, png See also: Annotations for §8.10 and Ch.8

The inequalities in (8.10.1) and (8.10.2) are reversed when $a\geq 1$ . If $\vartheta$ is defined by

8.10.3		$x^{1-a}e^{x}\Gamma\left(a,x\right)=1+\frac{a-1}{x}\vartheta,$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $x$ : real variable, $a$ : parameter and $\vartheta$ Permalink: http://dlmf.nist.gov/8.10.E3 Encodings: TeX, pMML, png See also: Annotations for §8.10 and Ch.8

then $\vartheta\to 1$ as $x\to\infty$ , and

8.10.4		$0<\vartheta\leq 1,$
$x>0$ , $a\leq 2$ .
ⓘ Symbols: $x$ : real variable, $a$ : parameter and $\vartheta$ Referenced by: §8.10 Permalink: http://dlmf.nist.gov/8.10.E4 Encodings: TeX, pMML, png See also: Annotations for §8.10 and Ch.8

For further inequalities of these types see Qi and Mei (1999) and Neuman (2013).

Padé Approximants

ⓘ

Keywords:: Padé approximant, incomplete gamma functions
See also:: Annotations for §8.10 and Ch.8

For $n=1,2,\dots$ ,

8.10.5		$A_{n}<x^{1-a}e^{x}\Gamma\left(a,x\right)<B_{n},$
$x>0$ , $a<1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $x$ : real variable, $a$ : parameter, $n$ : nonnegative integer, $A_{n}$ : minima and $B_{n}$ : maxima Permalink: http://dlmf.nist.gov/8.10.E5 Encodings: TeX, pMML, png See also: Annotations for §8.10, §8.10 and Ch.8

where

8.10.6	$\displaystyle A_{1}$	$\displaystyle=\frac{x}{x+1-a},$
	$\displaystyle B_{1}$	$\displaystyle=\frac{x+1}{x+2-a},$
	$\displaystyle A_{2}$	$\displaystyle=\frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)},$
	$\displaystyle B_{2}$	$\displaystyle=\frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3-a)}.$
ⓘ Symbols: $x$ : real variable, $a$ : parameter, $A_{n}$ : minima and $B_{n}$ : maxima Permalink: http://dlmf.nist.gov/8.10.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §8.10, §8.10 and Ch.8

For hypergeometric polynomial representations of $A_{n}$ and $B_{n}$ , see Luke (1969b, §14.6).

Next, define

8.10.7		$I=\int_{0}^{x}t^{a-1}e^{t}\,\mathrm{d}t=\Gamma\left(a\right)x^{a}\gamma^{*}% \left(a,-x\right),$
$\Re a>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $x$ : real variable, $a$ : parameter and $I$ : integral Permalink: http://dlmf.nist.gov/8.10.E7 Encodings: TeX, pMML, png See also: Annotations for §8.10, §8.10 and Ch.8

Then

8.10.8		$\frac{(a+1)(a+2)-x}{(a+1)(a+2+x)}<ax^{-a}e^{-x}I<\frac{a+1}{a+1+x},$
$x>0$ , $a\geq 0$ .
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $x$ : real variable, $a$ : parameter and $I$ : integral Permalink: http://dlmf.nist.gov/8.10.E8 Encodings: TeX, pMML, png See also: Annotations for §8.10, §8.10 and Ch.8

Also, define

8.10.9	$\displaystyle c_{a}$	$\displaystyle=(\Gamma\left(1+a\right))^{1/(a-1)},$
8.10.9	$\displaystyle d_{a}$	$\displaystyle=(\Gamma\left(1+a\right))^{-1/a}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $a$ : parameter, $c_{a}$ : coefficients and $d_{a}$ : coefficients Permalink: http://dlmf.nist.gov/8.10.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.10, §8.10 and Ch.8

Then

8.10.10		$\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right)<x^{1-a}e^{x}\Gamma% \left(a,x\right)\leq\frac{x}{ac_{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1% \right),$
$x\geq 0$ , $0<a<1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $x$ : real variable, $a$ : parameter and $c_{a}$ : coefficients Referenced by: §8.10 Permalink: http://dlmf.nist.gov/8.10.E10 Encodings: TeX, pMML, png See also: Annotations for §8.10, §8.10 and Ch.8

and

8.10.11		$(1-e^{-\alpha_{a}x})^{a}\leq P\left(a,x\right)\leq(1-e^{-\beta_{a}x})^{a},$
$x\geq 0$ , $a>0$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $x$ : real variable, $a$ : parameter, $\alpha_{a}$ and $\beta_{a}$ Referenced by: (7.8.8), §8.10 Permalink: http://dlmf.nist.gov/8.10.E11 Encodings: TeX, pMML, png See also: Annotations for §8.10, §8.10 and Ch.8

where

8.10.12	$\displaystyle\alpha_{a}$	$\displaystyle=\begin{cases}1,&0<a<1,\\ d_{a},&a>1,\end{cases}$
8.10.12	$\displaystyle\beta_{a}$	$\displaystyle=\begin{cases}d_{a},&0<a<1,\\ 1,&a>1.\end{cases}$
ⓘ Symbols: $a$ : parameter, $d_{a}$ : coefficients, $\alpha_{a}$ and $\beta_{a}$ Permalink: http://dlmf.nist.gov/8.10.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.10, §8.10 and Ch.8

Equalities in (8.10.11) apply only when $a=1$ .

Lastly,

8.10.13		$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma% \left(n,n-1\right)}{\Gamma\left(n\right)},$
$n=1,2,3,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $n$ : nonnegative integer Referenced by: §8.10 Permalink: http://dlmf.nist.gov/8.10.E13 Encodings: TeX, pMML, png See also: Annotations for §8.10, §8.10 and Ch.8

8.9 Continued Fractions 8.11 Asymptotic Approximations and Expansions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§8.10 Inequalities

Padé Approximants

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.