This seems to be used in the proof of Proposition 5.4 in "Analytic Number Theory" by Ivaniec and Kowalski. It's about estimating a quotient of Gamma functions. In the proof it says, that for fixed $ \Re(s) = \sigma > 0, \Re(u) = \beta > - \sigma $ we have by Stirlings formula $$ \frac{\Gamma(s + u)}{\Gamma(s)} \ll \frac{|s+u|^{\sigma + \beta - 1/2}}{|s|^{\sigma-1/2}} \exp \Big(\frac{\pi}{2}(|s|-|s+u|) \Big) \\ \ll (|s|+3)^\beta \exp (\frac{\pi}{2}|u|).$$
The first inequality is just an application of Stirlings formula, this is clear to me. But I can't derive the second inequality. It seems to me, that the $ \exp $-term is just estimated by the reversed triangle inequality. But what happens with the quotient then?
I'm also not completly sure, if this is an estimation with respect to $s$ AND $u$, or only with respect to $u$ while $s$ is assumed to be constant. I'd say it has to be with respect to both. Either way, I don't make progress on the inequality. It may also tacitly be assumed that $ \sigma < \frac{1}{2} $, but this doesn't help me either.
Also the resulting factor $ (|s|+3)^\beta $ contributes thereafter in the proof to the analytic conductor. So the estimation of the quotient probably looks different first, and is then estimated by the specific term $ (|s|+3)^\beta $.