Jensen's Inequality

Jensen's inequality is often formulated in probabilistic terms, using expected values. It is a direct reformulation of the definition of a concave function.

Theorem: Suppose $g$ is a concave function on $\mathcal{S}\subset\mathbb{R}^n,$ and suppose $\pi$ is a weight function such that $\int_\mathcal{S}\pi(x)dx=1,$ and $\mu\mathop{=}\limits^{\Delta}\int_\mathcal{S} x\pi(x)dx$ is finite. Then $\int_\mathcal{S}\pi(x)g(x)dx\leq g(\mu),$ with equality if and only if $g$ is linear a.e.

Proof: If $g$ is concave, then $g(x)\leq g(\mu)+(x-\mu)'\eta(\mu),$ where $\eta(\mu)$ is an arbitrary element of the subgradient of $g$ at $\mu.$ Multiplying both sides by $\pi(x),$ and integrating gives the required result. QED