In any hypothesis test, we have a false positive rate (a.k.a. $\alpha$; a.k.a. type 1 error rate) and false negative rate (a.k.a. $\beta$; a.k.a. type 2 error rate). It's clear that we can set the $\alpha$ to whatever we like. But the smaller the $\alpha$, the larger the $\beta$ so there is a tradeoff. If we consider some effect size (difference between modes of yellow and purple curve in the figure below), the distribution of the statistic under the null hypothesis shown in the yellow curve and the distribution of the alternate hypothesis shown in the purple curve, this tradeoff looks something like this:
Now my question: All such $\alpha$-$\beta$ curve's I've seen are decreasing (of course) but also convex as long as the distributions of the null and alternate hypothesis are the same but simply shifted. This implies that we can never have one test that has a lower $\beta$ for some values of $\alpha$, but a higher one for other values. I'm looking for a proof of this claim or the outline of one.