§8.10 Inequalities

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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

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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