Let $X$ and $Y$ be two zero-mean independent and identically distributed random variables. Then, is there any 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, respectively.

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?

Let $X$ and $Y$ be two independent and identically distributed random variables. Then, is there any bound for the following ratio? $$\frac{\mathbb{E}[|X+Y|]}{\mathbb{E}[|X|+|Y|]}$$ where $\mathbb{E}$ and $|.|$ are the expectation and absolute value, respectively.

Let $X$ and $Y$ be two zero-mean independent and identically distributed random variables. Then, is there any 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, respectively.