Is there a simple upper bound of the following fraction of gamma functions for any $a,b\geq1/2$: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$ An upper bound in the following form is enough: there exists a constant $C>0$ and a function $f(a,b)$ such that: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$

I have a guess of $f$, which is $\frac{a+b}{a}$. Is it true?