TITLE: Prove an inequality similar to Jensen's inequality
QUESTION [5 upvotes]: Prove that 
$$\mathbb{E}(\sqrt{X})\leq\sqrt{\mathbb{E}Y}$$
where random variables $X,Y>0$ and $\mathbb{E}\left[\frac{X}{Y}\right]\leq1.$

My attempt:
This looks very much like Jensen's inequality.
According to Jensen, $\mathbb{E}(\sqrt{X})\leq\sqrt{\mathbb{E}X},$ then the desired inequality is true if we can prove $\mathbb{E}X\leq\mathbb{E}Y.$ But this seems impossible to prove because $X$ and $Y$ are not independent.
Any help will be appreciated!

REPLY [7 votes]: The Cauchy-Schwarz inequality gives:
$$\mathbb{E}[\sqrt{X}]=\mathbb{E}\left[\sqrt{Y}\cdot\sqrt{\frac{X}{Y}}\right]\leq\sqrt{\mathbb{E}[Y]\cdot\mathbb{E}\left[\frac{X}{Y}\right]}\leq\sqrt{\mathbb{E}[Y]}.$$