Let $m>0$, $r>0$ and $0< k < m$ be three positive real numbers. I want to find a tight lower bound of the special case of the Beta function, which is given by $$ B\left(\frac{m-k}{2}, \frac{r+k}{2}\right) = \frac{\Gamma\left(\frac{m-k}{2}\right)\Gamma\left(\frac{r+k}{2}\right)}{\Gamma\left(\frac{m+r}{2}\right)}, $$ where $\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d}t$ denotes the Gamma function.
Could someone share me some closely related references?