# §8.10 Inequalities

 8.10.1 $x^{1-a}e^{x}\Gamma\left(a,x\right)\leq 1,$ $x>0$, $0, ⓘ 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 8
 8.10.2 $\gamma\left(a,x\right)\geq\frac{x^{a-1}}{a}(1-e^{-x}),$ $x>0$, $0. ⓘ 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 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 $x^{1-a}e^{x}\Gamma\left(a,x\right)=1+\frac{a-1}{x}\vartheta,$

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 8

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

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

 8.10.5 $A_{n} $x>0$, $a<1$,

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 8

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

Next, define

 8.10.7 $I=\int_{0}^{x}t^{a-1}e^{t}\mathrm{d}t=\Gamma\left(a\right)x^{a}\gamma^{*}\left% (a,-x\right),$ $\Re a>0$.

Then

 8.10.8 $\frac{(a+1)(a+2)-x}{(a+1)(a+2+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 8

Also, define

 8.10.9 $\displaystyle c_{a}$ $\displaystyle=(\Gamma\left(1+a\right))^{1/(a-1)},$ $\displaystyle d_{a}$ $\displaystyle=(\Gamma\left(1+a\right))^{-1/a}.$ ⓘ 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 8

Then

 8.10.10 $\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right) $x\geq 0$, $0, ⓘ 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 8

and

 8.10.11 $(1-e^{-\alpha_{a}x})^{a}\leq P\left(a,x\right)\leq(1-e^{-\beta_{a}x})^{a},$ $x\geq 0$, $a>0$,

where

 8.10.12 $\displaystyle\alpha_{a}$ $\displaystyle=\begin{cases}1,&01,\end{cases}$ $\displaystyle\beta_{a}$ $\displaystyle=\begin{cases}d_{a},&01.\end{cases}$ ⓘ 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 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 8