Several Colorful Inequalities

For a positive random variable X, let μα(X) be the αth moment of X. Mark Brown proves that for positive and independent random variables X, Y, F(X + Y) ≥ F(X) + F(Y), where F (X) is the ratio μ-1 over μ-2. We prove this inequality and several generalizations by a method which can be used to prove the Schwarz inequality, but which is not widely appreciated.