For any constant $a>0$ and strictly increasing function $h$ s.t. $h(a)>0$. $P\{Y \geq a\} \leq \frac{E[h(Y)]}{h(a)} $

Question

For any constant $a>0$ and strictly increasing function $h$ s.t. $h(a)>0$. $P\{Y \geq a\} \leq \frac{E[h(Y)]}{h(a)} $

I am having trouble figuring out how to start this problem. I was thinking to start with $$P\{Y \geq a\} = 1 - P\{Y < a\} $$ and the definition of expectation on $E[h(Y)]$. But do not know how to go from there or whether this approach is right.

Answer 1

Note that $Y \geq a \Longleftrightarrow h(Y) \geq h(a)$ since $h$ is increasing. Then apply Markov's inequality.