Jensen's Inequality [lcc-002R]

May 19, 2024
Eigil Fjeldgren Rischel

The inequality $f(\theta x + (1-\theta )x') \leq \theta f(x) + (1-\theta )f(x'),$ which holds whenever $f$ is convex, is called Jensen's inequality. Sometimes this name is used for a stronger version of this inequality, like the claim that $f(\mathbb {E} X) \leq \mathbb {E} f(X)$ if $X$ is a random variable valued in the domain of $f$ . These generally follow just from convexity of $f$ .