I'm trying to find an upper bound for
$$\left|\frac{\Gamma(z_1+n)}{\Gamma(z_2+n)}\right|$$
with $n\in\mathbb{N}$ and $z_1,z_2\in\mathbb{C}$. It's equivalent to find an upper and lower bound for $\displaystyle \left|(z)_n\right|=\left|\frac{\Gamma(z+n)}{\Gamma(z)}\right|$ with $(-)_n$ pochhammer symbol.
Another way may be look for an upper and lower boun for $\displaystyle \left|\Gamma(z+n)\right|$.
Any help or references are wellcome.