The answer to another question (Upper bound of the fraction of Gamma functions) gave an asymptotic upper bound for an expression with Gamma functions: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\!\leq \,C\,\frac{a+b}a, \forall a,b\geq\frac12$$
What is the best possible value for the constant $C$ in that statement?
The optimal $C$ is $\mathrm{e}$.
Proof:
We have $$\ln C \ge \ln a - \ln(a + b) + \frac{\ln \Gamma(a + b) -\ln a - \ln\Gamma(a) - \ln\Gamma(b)}{a}.$$
Let $$F(a, b) := \ln a - \ln(a + b) + \frac{\ln \Gamma(a + b) -\ln a - \ln\Gamma(a) - \ln\Gamma(b)}{a}.$$ We have $$\frac{\partial F}{\partial b} = - \frac{1}{a+b} + \frac{\psi(a + b) - \psi(b)}{a} \ge 0 \tag{1}$$ where $\psi(\cdot)$ is the digamma function defined by $\psi(u) = \frac{\mathrm{d} \ln \Gamma(u)}{\mathrm{d} u} = \frac{\Gamma'(u)}{\Gamma(u)}$. The proof of (1) is given at the end.
Fixed $a\ge 1/2$, we have $$G(a) := \lim_{b\to \infty} F(a, b) = \ln a + \frac{ -\ln a - \ln\Gamma(a)}{a} $$ where we have used $$\lim_{b\to \infty} -\ln(a + b) + \frac{\ln \Gamma(a + b) - \ln\Gamma(b)}{a} = 0.$$ (Note: Use $\sqrt{2\pi}\, x^{x-1/2}\mathrm{e}^{-x} \le \Gamma(x) \le \sqrt{2\pi}\, x^{x-1/2}\mathrm{e}^{-x}\mathrm{e}^{\frac{1}{12x}}$ for all $x > 0$.)
We have $$ a^2 G'(a) = a - 1 - a\psi(a) + \ln a + \ln \Gamma(a) \ge 0.\tag{2} $$ The proof of (2) is given at the end.
We have $$\lim_{a\to \infty} G(a) = 1.$$ (Note: Use $\sqrt{2\pi}\, x^{x-1/2}\mathrm{e}^{-x} \le \Gamma(x) \le \sqrt{2\pi}\, x^{x-1/2}\mathrm{e}^{-x}\mathrm{e}^{\frac{1}{12x}}$ for all $x > 0$.)
Thus, the optimal $C$ is $\mathrm{e}$.
Proof of (1):
Using Theorem 5 in [1]: for all $u > 0$, $$\ln u - \frac{1}{2u} - \frac{1}{12u^2} < \psi(u) < \ln u - \frac{1}{2u} - \frac{1}{12(u+1/14)^2},$$ we have \begin{align*} &- \frac{1}{a+b} + \frac{\psi(a + b) - \psi(b)}{a}\\ \ge{}&- \frac{1}{a+b} + \frac{1}{a} \left(\ln (a+b) - \frac{1}{2(a+b)} - \frac{1}{12(a+b)^2}\right)\\ &\qquad - \frac{1}{a}\left(\ln b - \frac{1}{2b} - \frac{1}{12(b+1/14)^2}\right)\\ ={}& \frac{1}{a}\ln(1 + a/b) - \frac{1}{a+b} - \frac{1}{2a(a+b)} - \frac{1}{12a(a+b)^2} + \frac{1}{2ab} + \frac{1}{12a(b+1/14)^2}\\ \ge{}& \frac{1}{a}\left(\ln(1 + a/b) - \frac{a/b}{1 + a/b}\right) + \frac{1}{2ab} - \frac{1}{2a(a+b)} + \frac{1}{12a(b+1/14)^2} - \frac{1}{12a(a+b)^2}\\ \ge{}&0 \end{align*} where we use $\ln(1+x) \ge \frac{x}{1+x}$ for all $x \ge 0$.
We are done.
$\phantom{2}$
Proof of (2):
Using $\Gamma(x) \ge \sqrt{2\pi}\, x^{x-1/2}\mathrm{e}^{-x}$ and $\psi(u) < \ln u - \frac{1}{2u} - \frac{1}{12(u+1/14)^2}$ for all $u > 0$ (Theorem 5 in [1]), we have \begin{align*} &a - 1 - a\psi(a) + \ln a + \ln \Gamma(a)\\ \ge{}& a - 1 - a \left(\ln a - \frac{1}{2a} - \frac{1}{12(a+1/14)^2}\right) + \ln a + \frac12\ln(2\pi) + (a-1/2)\ln a - a\\ ={}& \frac12\ln(2\pi a) - \frac12 + \frac{a}{12(a+1/14)^2}\\ \ge{}& 0. \end{align*}
We are done.
Reference
[1] L. Gordon, “A stochastic approach to the gamma function”, Amer. Math. Monthly, 9(101), 1994, 858-865.
Too long for a comment. You know certainly that the so-called Beta function $B$ ( a classical special function) is defined by $$ B(a,b)=\frac{\Gamma (a)\Gamma (b)}{\Gamma (a+b)}=\int_0^1 t^{a-1}(1-t)^{b-1} dt \quad\text{ for $\Re a>0, \Re b> 0$.} $$ You are thus looking for a lowerbound for the $B$ function. When $a, b$ are real-valued and large the Beta function is equivalent to $$ \sqrt{2π}\frac{a^{a-\frac 12}b^{b-\frac 12}}{(a+b)^{a+b-\frac 12}}. $$ You can also fix $b$ and consider $a$ large to get the equivalent $$ \Gamma (b) a^{-b}. $$ Both asymptotics are obtained from Stirling's formula.
Taking $f(a,b)=\frac{a+b}{a}$ as in the other question, I conjecture that the optimal $C$ is $e$.
One bit of evidence is that the limit of the ratio for $a=cb$ and $b\to\infty$ is $(c+1)^{1/c}$ which converges to $e$ as $c\to\infty$. Also the limit for $a=b^{1/2}$ as $b\to\infty$ is exactly $e$ and the ratio seems to be increasing.
