MathDB

Let

f,g,h : \mathbb R \to \mathbb R

be continuous functions and

X

be a random variable such that

E(g(X)h(X))=0

and

E(g(X)^2) \neq 0 \neq E(h(X)^2)

. Prove that

E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}.

You may assume that all expected values exist.
Proposed by Cristi Calin

Problem Statement