A simple upper bound for the Gamma function of a product

Question

Given integer $p \ge 1$ and $q \in [0,1)$, is there a simple upper bound on the quantity $\Gamma(p(1-q))$, where $\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt$ is the gamma function?

Answer 1

If $p(1-q)\ge 1$, then since $\Gamma$ is increasing on $[2,\infty)$, then the upper bound is $\Gamma(p)=p!$

If $0<p(1-q)<1$, then (from integral by parts) $\Gamma(s+1) = s\Gamma(s)$, so $$\Gamma(p(1-q)) = \frac{1}{p(1-q)}\Gamma(p(1-q)+1)\le \frac{2}{p(1-q)}.$$

So I guess a very rough upper limit is $\displaystyle\max\left\{p!, \frac{2}{p(1-q)}\right\}$, and you can discard the max sign if you know the range of $p(1-q)$.