Equality in Jensen's inequality

ProofBeginner⏱ 15 minFoundationsjensenconvexityequality case

Problem

Let $\varphi$ be strictly convex and $X$ integrable with $\E|\varphi(X)|<\infty$ . Prove that $\varphi(\E X) = \E[\varphi(X)]$ if and only if $X$ is almost surely constant. Then use Jensen to show that for a positive random variable, $\E[1/X] \ge 1/\E[X]$ , and characterise equality.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

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Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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