In the question : Upper bound of the fraction of Gamma functions 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 $C$?