Suppose $X$ and $Y$ are given random variables with
a) $\mathbb{E}( X + Y ) = \mathbb{E}( X - Y ) = 0$
b) $\text{Var}( X + Y ) = 3$
c) $\text{Var}( X - Y ) = 1$
We have to show $\mathbb{E}| X + Y |<\sqrt{3}$
Now I tried and got $|\mathbb{E}( X + Y) |<\sqrt{3}$ but couldn't establish the relationship using Jensen's inequality. Please help.
Using the definition of variance
$$Var(X+Y)=\mathbb{E}((X+Y)^2)-\mathbb{E}(X+Y)^2=\mathbb{E}((X+Y)^2),$$
where the last equality follows from a). On the other hand then by b)
$$\mathbb{E}[(X+Y)^2]=\mathbb{E}[|X+Y|^2]=3$$
But because $z^2$ is a convex function, by Jensen's inequality
$$3=\mathbb{E}[|X+Y|^2]>\mathbb{E}[|X+Y|]^2$$
Therefore taking square roots and given the positivity of $\mathbb{E}[|X+Y|]$, we obtain
$$\mathbb{E}[|X+Y|]<\sqrt{3}.$$