= Strict Jensen equality criterion
{title2=$\mathbb E[f(X)]=f(\mathbb EX)\iff X=\mathbb EX\text{ a.s.}$}
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 $m=\mathbb EX$ 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 $m$, so
$$
h(t)=f(t)-f(m)-f'(m)(t-m)\geq0.
$$
Strict <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>. Therefore
$$
\mathbb E[f(X)]-f(m)=\mathbb E[h(X)]\geq0,
$$
and 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.
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