Reference Request: Probability-and-Convexity Statement

I can prove the following statement, but it would be even better to have a reference I can cite. Does anyone know where this (or a generalization) is proven?

Proposition. Let $0 \leq b \leq a$, and for a random variable $X$ supported on $[0,a]$, let$$\mathbb{E}_b[X] = \mathbb{E}[X \mathbb{1}_{X \leq b}].$$Suppose $X$ is such a random variable with$$(*) \ \ \mathbb{E}[X] \geq \alpha \text{ and } \mathbb{E}_b[X] \geq \beta,$$where $0 \leq \beta \leq \alpha$. Then the random variable $Z$ with$$\mathbb{P}(Z=b) = \beta/b, \mathbb{P}(Z=a) = (\alpha-\beta)/a, \text{ and } \mathbb{P}(Z=0) = 1 - (\mathbb{P}(Z=b) + \mathbb{P}(Z=a)),$$satisfies $(*)$ (with equality) and, for any convex $g : [0,a] \to \mathbb{R}$ with $g(a) \leq g(0)$,$$\mathbb{E}[g(Z)] \geq \mathbb{E}[g(X)].$$