When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

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.

• Equality may be possible but I doubt if strict inequality is true for any positive $a_i, b_i$, you may be better off posting the actual expression you want maximised. – Macavity May 1 '16 at 7:29
• Strict inequality is true for many values. For example, $\beta(35+50,46+68)> \beta(35, 46)\beta(50, 68)$. – Kortlek May 2 '16 at 19:19
• @Macavity In fact, experimentation shows it is only true when $\frac{a_1}{b_1}\approx \frac{a_2}{b_2}$. I don't have an explicit expression. I'm using an algorithm for finding an optimal clustering of data $(a_i, b_i)$ and this algorithm requieres you to sort the data in a specific way. So for example, guided by my experimentation, sorting the data so that $\frac{a_i}{b_i} \le \frac{a_{i+1}}{b_{i+1}}$, I very often find the optimal clustering, but not always. Therefore I wanted to find some more refined condition, but I'm starting to suspect I will have to settle for this. – Kortlek May 2 '16 at 19:38
• you could use upper and lower bounds on the Gamma function to try and achieve some bounds on the Beta function but nothing looks promising... – Jack Tiger Lam May 7 '16 at 2:19