$a,b>0$, we have the estimate: $$C(b)^{-1}(a+1)^b\leq \frac{\Gamma(a+b)}{\Gamma(a)}\leq C(b)(a+1)^b$$ Where $C(b)$ is a constant does not depend on $a$.
Actually, for $b\geq 1$, this can be easily proved by using the formulas $\Gamma(x+1)=x\Gamma(x)$. But how to prove the case of $0<b<1$? Thanks very much.