from $E[(\phi(X)^{k})^{\frac{s}{k}}]$ to $E[\phi(X)^{k}]^{\frac{s}{k}}$

Question

Is there any sort of inequality or rule to get some relation between

$E[(\phi(X)^{k})^{\frac{s}{k}}]$ and $E[\phi(X)^{k}]^{\frac{s}{k}}$

Where $s$ and $k$ are integers and $s \in O(\log k)$. Further $\phi$ is a continous function.

Answer 1

At least for large enough $k$, you know that $s \leq k$ since $s \in O(\log k)$. This means that $x \mapsto x^{\frac{k}{s}}$ is convex and hence you can apply Jensen's inequality to the random variable $\phi(X)^s$ to see that $$\mathbb{E}[\phi(X)^s]^{\frac{k}{s}} \leq \mathbb{E}[\phi(X)^k]$$ Raising this to the power $\frac{s}{k}$ gives an inequality between the desired terms (for $k$ sufficiently large).