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Suppose that the expected value of a random variable with pdf fL(x) is greater than the expected value of a random variable with pdf fH(x).

Is it true that the integral over the common support INT{ (fH(x)^2)/fL(x) } dx is larger than one? Is it true if the CDF FH(x)<=FL(X)?

I've checked family of distributions x^a, exponential, Normal, binomial and the claim is true in these classes, but I dont know if its true in general....