Say $X$ is a bounded random variable with mean $0$, and $f$ is a convex function with $f(0)=0$. Then the usual Jensen's inequality states $$0≤\mathbb Ef(X).$$ Nothing special. However, if I add the condition that $\mathbb P\{X≥a\}≥b$ for some positive $a,b$, then the equality cannot hold. And I claim that $$0<bf(a)+(1-b)f(-ab/(1-b))≤\mathbb Ef(X).$$ In other words, I claim that the worst case scenario is when all masses concentrate on $a$ and on $-ab/(1-b)$. (The number $-ab/(1-b)$ is such that the mean is still $0$.)
Question: Is it (trivially) true? If so, is there a name or something I can cite?