In the question : https://mathoverflow.net/q/313103/125166 the asymptotic upper bound for the fraction of Gamma functions have been established:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$. 

What is a non-asymptotic bound for this ratio, i.e. what is the constant $C$?