Consider the following concentration inequality.
The problem is :
The inequalities are easily proven with the help of Markov's inequality. But I cannot find a strategy to construct an example to show that it cannot be improved. My rough strategy is : For every $\epsilon>0$, to construct a random variable $X_{\epsilon}$ with $E(X_{\epsilon})=0$ and $var(X_{\epsilon})=\sigma^2$ such that $P(X_{\epsilon} \geq x)>\frac{\sigma^2}{\sigma^2+x^2}$ if $x>0$ and $P(X_{\epsilon} \geq x)<\frac{x^2}{\sigma^2+x^2}$ if $x<0$. How can I construct such $X_{\epsilon}$? Thanks in advance.
Source : Rohatgi-Saleh, p.$98$, problem $3$.
Why not let $X\equiv 0$? That achieves equality in both cases.