Digital Library of Mathematical Functions
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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.10 Inequalities

8.10.1 x1-axΓ(a,x)1,
x>0, 0<a1,
8.10.2 γ(a,x)xa-1a(1--x),
x>0, 0<a1.

The inequalities in (8.10.1) and (8.10.2) are reversed when a1. If ϑ is defined by

8.10.3 x1-axΓ(a,x)=1+a-1xϑ,

then ϑ1 as x, and

8.10.4 0<ϑ1,
x>0, a2.

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

Padé Approximants

For n=1,2,,

8.10.5 An<x1-axΓ(a,x)<Bn,
x>0, a<1,

where

8.10.6 A1 =xx+1-a,
B1 =x+1x+2-a,
A2 =x(x+3-a)x2+2(2-a)x+(1-a)(2-a),
B2 =x2+(5-a)x+2x2+2(3-a)x+(2-a)(3-a).

For hypergeometric polynomial representations of An and Bn, see Luke (1969b, §14.6).

Next, define

8.10.7 I=0xta-1tt=Γ(a)xaγ*(a,-x),
a>0.

Then

8.10.8 (a+1)(a+2)-x(a+1)(a+2+x)<ax-a-xI<a+1a+1+x,
x>0, a0.

Also, define

8.10.9 ca =(Γ(1+a))1/(a-1),
da =(Γ(1+a))-1/a.

Then

8.10.10 x2a((1+2x)a-1)<x1-axΓ(a,x)xaca((1+cax)a-1),
x0, 0<a<1,

and

8.10.11 (1--αax)aP(a,x)(1--βax)a,
x0, a>0,

where

8.10.12 αa ={1,0<a<1,da,a>1,
βa ={da,0<a<1,1,a>1.

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

Lastly,

8.10.13 Γ(n,n)Γ(n)<12<Γ(n,n-1)Γ(n),
n=1,2,3,.