I'm wondering if there is any non-asymptotic upper bound for the following Gamma function:
$$f\_a(x)=\int\_{x}^{\infty}t^a\exp(-t)dt$$
for $x>0,a>0$? Something like $x^a\exp(-x)$?



Non-asymptotic upper bound of right tail of Gamma function
In Gabcke thesis, that you can download at 
<http://hdl.handle.net/11858/00-1735-0000-0022-6013-8>
page 84 you can find 


Theorem: The incomplete gamma function 
$$\Gamma(a,x)=\int\_x^\infty e^{-v}v^{a-1}\,dv,\qquad \text{$x>0$ and $a$ real numbers}$$
satisfies for $a\ge 1$ and $x>a$  

$$\Gamma(a,x)\le a e^{-x} x^{a-1}.$$


