TITLE: Is there a universal bound for this ratio of expectations?
QUESTION [6 upvotes]: Let $X$ and $Y$ be two zero-mean independent and identically distributed random variables.  Is there a bound for the following ratio,
$$\frac{\mathbb{E}[|X+Y|]}{\mathbb{E}[|X|+|Y|]}=\frac{\mathbb{E}[|X+Y|]}{2\mathbb{E}[|X|]} ,$$
where $\mathbb{E}$ and $|.|$ are the expectation and absolute value operations, respectively?

REPLY [5 votes]: This is Jensen's inequality.  Applied to the distribution of Y one gets $E|X+Y| \ge |X + E(Y)|$