I am looking for an upper bound on the following ratio of Gamma functions \begin{align} \frac{\Gamma \left( \frac{n}{2}+x\right)}{\Gamma \left( \frac{n}{2}\right)} \end{align}
where $x \ge 0$ (real) and $n\ge 1$ (integer).
Here is the bound I was able to come up with \begin{align} \frac{\Gamma \left( \frac{n}{2}+x\right)}{\Gamma \left( \frac{n}{2}\right)} \le \frac{\Gamma \left( \left \lceil \frac{n}{2}+x \right \rceil \right) }{\Gamma \left( \left \lfloor \frac{n}{2} \right \rfloor \right)}=\frac{\left( \left \lceil \frac{n}{2}+x \right \rceil -1 \right) !}{\left( \left \lfloor \frac{n}{2} \right \rfloor -1\right)!} \end{align}