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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.

Thank you.

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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).

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