Suppose $s, w$ are complex numbers with positive real part. I have come across a particular bound that I have seen multiple times, but which I do not know how to prove:

$$\frac{\Gamma(s+u)}{\Gamma(s)} \ll (3 + \lvert s \rvert)^{\text{Re} \,u}e^{\pi\lvert u \rvert / 2},$$

where the implicit constant should depend on the real parts of $s$ and $u$ in a way that I would like to understand better. It's clear that this should be approached with Stirling's formula. One quick application of Stirling's formula leads to a possible intermediary step

$$\frac{\Gamma(s+u)}{\Gamma(s)} \ll \frac{\lvert s + u \rvert^{\text{Re}(s + w) - \frac 12}}{\lvert s \rvert^{\text{Re}(s) - \frac 12}} \exp\left( \frac{\pi}{2} (\lvert s \rvert - \lvert s + u \rvert)\right).$$

There should be a reasonable and mostly explicit way of getting from here to the desired inequality, and with explicit understanding of the dependence on the real parts of $s$ and $u$.


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