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?

upperbound, better than $1$, depending on some moments of $X$. $\endgroup$