Applying Jensen's inequality to find a bound

Question

I'm not sure how to begin with this question, apart from writing some basic relationships based on Jensen's inequality. Any suggestions would be appreciated.

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.

Attempt at a solution:

E[|X|$^α$] $\ge$ (E[|X|])$^α$
E[|X|$^β$] $\ge$ (E[|X|])$^β$

f(E[|X|$^α$]) $\le$ E[f(|X|$^α$)]
f(E[|X|$^β$]) $\le$ E[f(|X|$^β$)]

Answer 1

$E|X|^{\alpha}\leq (E|X|^{\beta})^{\alpha /\beta}$ by Jensen's inequality applied to the convex function $g(t)= t^{\beta /\alpha }$. Indeed $(E|X|^{\alpha})^{\beta /\alpha} =g(E|X|^{\alpha})) \leq (Eg(|X|^{\alpha}))$ which gives the inequality. Hence we can take $f(x)=x^{\alpha /\beta}$.