Bounding Mean Absolute Deviation from the Mean using Variance
Let $X$ be a real random variable with $E[X] = 0$. Let $a = E[|X|]$ and $b = Var(X)$.
(a) Find the best possible upper bound on $a$ in terms of $b$.
(b) Find the best possible upper bound on $b$ in terms of $a$.
(c) Demonstrate that your bounds are tight by constructing a random variable $X_{a,b}$ for each pair
$(a, b)$ satisfying the inequalities from the previous parts.
Attempt at a solution:
By the definition of $var(X)$, $$b = E[X^2] - (E[X])^2$$ Since $E[X] = 0$, this reduces to $$b = E[X^2]$$ By Jensen's Inequality, $$b \ge (E[|X|])^2$$ $E[|X|] = a$; therefore $$b \ge a^2$$ $a\le\sqrt b$ which gives a best possible upper bound on $a$ in terms of $b$.
However, I am unsure how to find a best possible upper bound on $b$ in terms of $a$. Any help would be appreciated. Thanks!