Strict Jensen equality criterion ($\mathbb{E}[f(X)]=f(\mathbb{E}X)⟺X=\mathbb{E}X\text{ a.s.}$)

Let $X$ be an integrable random variable taking values in an open interval $I$. Suppose $f:I\to \mathbb{R}$ is a differentiable strictly convex function. Then its expected value has equality in Jensen's inequality exactly when $X$ is constant almost surely.

First $m=\mathbb{E}X$ lies in $I$: at each finite endpoint, taking the expected value of the strictly positive distance to that endpoint gives a strictly positive distance for $m$. A tangent affine function supports $f$ at $m$, soStrict convexity forces $h(t)>0$ for $t\ne m$. Indeed, equality at another point would force the graph to coincide with its supporting line on the connecting line segment, contradicting strict convexity. Thereforeand the right side is zero exactly when $X=m$ almost surely.

The tangent affine function of $X$ is an integrable random variable, so $f(X)$ has an integrable negative part. Thus $\mathbb{E}[f(X)]$ is well defined, possibly $+\infty$, even if absolute integrability was not assumed. Equality to $f(m)$ automatically excludes that infinite case.