Let $x, y \geq 1$ be two real numbers.
I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$?
Any references or ideas are very appreciated.
You might be interested in the Beta function. In particular, we have the following relation: $$\Gamma(x)\Gamma(y)=B(x,y)\Gamma(x+y).$$ Hence, your problem reduces to finding bounds (or asymptotics) for the Beta function (see the section Approximations).