For positive random variables $X,Y$ prove that if $E(X/Y)<1$ then $E(\sqrt{X})\le \sqrt{E(Y)}$. I can use Jensen $E(\sqrt{X})\le \sqrt{E(X)} = \sqrt{E((X/Y)Y)}\le \sqrt{E(Y)}$. I am not sure about validity of the last step in my proof. Is this correct or how should I prove it?
2026-03-25 12:45:39.1774442739
Expected value inequality
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Take $\sqrt{X}=\sqrt{\frac{X}{Y}}\sqrt{Y}$. Now apply Holder inequality.