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Consider the regularized incomplete beta function $I_x(a, b)$ with $x \in [0,1]$ and $a, b > 0$. I am hypothesizing that the function is monotone decreasing with respect to $a$ and monotone increasing with respect to $b$, but I have had a hard time proving or disproving it. I believe this to be true because of the behavior of the beta distribution, and I haven't found any counterexamples from computation.

I noticed that this post addresses a similar problem, but I don't see how the method can be adapt to my problem directly, since the core construction is to find a change of variable so that the top and bottom integral are on the same interval.

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By looking through google results once more, I found the desired result in theorem 1 of this paper.

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