I am interested in providing an upper bound on the expression $\big(\Gamma(1+p/s)\big)^{1/p}$ where $p\geq 2$ and $0<s<1$. I want to prove that $\big(\Gamma(1+p/s)\big)^{1/p} \leq C_s p^{1/s}$ for some constant $C_s$ dependent solely on $s$.
I wish to utilize a bound from Stirling's approximation for factorials: $n!\leq en^{n+1/2}e^{-n}$. However, I'm not sure if it generalizes to the Gamma function. That is, I was wondering if we have $\Gamma(1+z) \leq e z^{z+1/2}e^{-z}$ for all $z\in\mathbb{C}$. I have not been able to find a reference for this.
Supposing this is true, we would then have $$\Gamma(1+p/s) \leq e (p/s)^{p/s}(p/s)^{1/2}e^{-p/s} \leq C_sp^{1/2}(p/s)^{p/s}.$$ Hence, we have (I might have to treat the cases where $C_s\leq 1$ differently) $$\big(\Gamma(1+p/s)\big)^{1/p} \leq C'_sp^{1/(2p)}p^{1/s} \leq C''_sp^{1/s}$$ since $p^{1/(2p)}$ is uniformly bounded for $p>0$.
Therefore, my question is whether the bound $\Gamma(1+z) \leq e z^{z+1/2}e^{-z}$ is true and, if yes, where can I find a proof or reference for it.