§8.10 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).
Keywords:: incomplete gamma functions, inequalities
Permalink:: http://dlmf.nist.gov/8.10

8.10.1 $x^{{1-a}}e^{x}\mathop{\Gamma\/}\nolimits\!\left(a,x\right)\leq 1,$ $x>0$ , $0<a\leq 1$ ,

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Referenced by:: §8.10
Permalink:: http://dlmf.nist.gov/8.10.E1
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8.10.2 $\mathop{\gamma\/}\nolimits\!\left(a,x\right)\geq\frac{x^{{a-1}}}{a}(1-e^{{-x}}),$ $x>0$ , $0<a\leq 1$ .

Symbols:: $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Referenced by:: §8.10
Permalink:: http://dlmf.nist.gov/8.10.E2
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The inequalities in (8.10.1) and (8.10.2) are reversed when $a\geq 1$ . If $\vartheta$ is defined by

8.10.3 $x^{{1-a}}e^{x}\mathop{\Gamma\/}\nolimits\!\left(a,x\right)=1+\frac{a-1}{x}\vartheta,$

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable, $a$ : parameter and $\vartheta$
Permalink:: http://dlmf.nist.gov/8.10.E3
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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 $\vartheta$
Permalink:: http://dlmf.nist.gov/8.10.E4
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For further inequalities of these types see Qi and Mei (1999).

¶ Padé Approximants

Keywords:: incomplete gamma functions

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

8.10.5 $A_{n}<x^{{1-a}}e^{x}\mathop{\Gamma\/}\nolimits\!\left(a,x\right)<B_{n},$ $x>0$ , $a<1$ ,

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : 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
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where

8.10.6

$A_{1}=\frac{x}{x+1-a},$

$B_{1}=\frac{x+1}{x+2-a},$

$A_{2}=\frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)},$

$B_{2}=\frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3-a)}.$

Symbols:: $x$ : real variable, $a$ : parameter, $A_{n}$ : minima and $B_{n}$ : maxima
Permalink:: http://dlmf.nist.gov/8.10.E6
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For hypergeometric polynomial representations of $A_{n}$ and $B_{n}$ , see Luke (1969b, §14.6).

Next, define

8.10.7 $I=\int _{0}^{x}t^{{a-1}}e^{t}dt=\mathop{\Gamma\/}\nolimits\!\left(a\right)x^{a}\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,-x\right),$ $\realpart{a}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable, $a$ : parameter and $I$ : integral
Permalink:: http://dlmf.nist.gov/8.10.E7
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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:: $e$ : base of exponential function, $x$ : real variable, $a$ : parameter and $I$ : integral
Permalink:: http://dlmf.nist.gov/8.10.E8
Encodings:: TeX, pMML, png

Also, define

8.10.9

$c_{a}=(\mathop{\Gamma\/}\nolimits\!\left(1+a\right))^{{1/(a-1)}},$

$d_{a}=(\mathop{\Gamma\/}\nolimits\!\left(1+a\right))^{{-1/a}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $a$ : parameter, $c_{a}$ : coefficients and $d_{a}$ : coefficients
Permalink:: http://dlmf.nist.gov/8.10.E9
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Then

8.10.10 $\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right)<x^{{1-a}}e^{x}\mathop{\Gamma\/}\nolimits\!\left(a,x\right)\leq\frac{x}{ac_{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right),$ $x\geq 0$ , $0<a<1$ ,

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable, $a$ : parameter and $c_{a}$ : coefficients
Referenced by:: §8.10
Permalink:: http://dlmf.nist.gov/8.10.E10
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and

8.10.11 $(1-e^{{-\alpha _{a}x}})^{a}\leq\mathop{P\/}\nolimits\!\left(a,x\right)\leq(1-e^{{-\beta _{a}x}})^{a},$ $x\geq 0$ , $a>0$ ,

Symbols:: $e$ : base of exponential function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $x$ : real variable, $a$ : parameter, $\alpha _{a}$ and $\beta _{a}$
Referenced by:: ¶ ‣ §8.10
Permalink:: http://dlmf.nist.gov/8.10.E11
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where

8.10.12

$\alpha _{a}=\begin{cases}1,&0<a<1,\\ d_{a},&a>1,\end{cases}$

$\beta _{a}=\begin{cases}d_{a},&0<a<1,\\ 1,&a>1.\end{cases}$

Symbols:: $a$ : parameter, $d_{a}$ : coefficients, $\alpha _{a}$ and $\beta _{a}$
Permalink:: http://dlmf.nist.gov/8.10.E12
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Equalities in (8.10.11) apply only when $a=1$ .

Lastly,

8.10.13 $\frac{\mathop{\Gamma\/}\nolimits\!\left(n,n\right)}{\mathop{\Gamma\/}\nolimits\!\left(n\right)}<\frac{1}{2}<\frac{\mathop{\Gamma\/}\nolimits\!\left(n,n-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(n\right)},$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $n$ : nonnegative integer
Referenced by:: §8.10
Permalink:: http://dlmf.nist.gov/8.10.E13
Encodings:: TeX, pMML, png