Jensen's Inequality

Question

For arbitrary α and β satisfying 0 < α ≤ β, use Jensen’s inequality to find a bound of the form
E[|X|$^α$] ≤ f(E[|X|$^β$]) for some function f.

Doesn't this bound follow naturally from the fact that α ≤ β without having to apply Jensen's inequality?

That is, since α ≤ β,

E[|X|$^α$] $\le$ E[|X|$^β$]

What am I missing here?

Answer 1

Hint:

Jensen's inequality tells you that if $\ f\ $ is convex then $\ f(E(Y))\le E(f(Y))\ $. If you take $\ Y=|X|^\alpha\ $, can you find a strictly increasing convex function $\ f\ $ such that $\ f(Y)= |X|^\beta\ $?