Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there exist some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)?$$
I'm working on maximizing an expression in which finding such a condition probably would be helpful, but I don't know how to go about finding them.
I would greatly appreciate some tips or some other sources that could be beneficial.