Inequalities between exponential means

Question

When considering a vector $\{x_1,\ldots,x_n\}$ the root-mean-power $$M_\alpha=(\frac{1}{n}\sum_{i=1}^n x_i^\alpha)^{\frac{1}{\alpha}}$$ for $\alpha$ a real number $\neq 0$ is known to be such that $$M_\alpha \leq M_\beta$$ for any $\beta > \alpha$, with equality if and only if $x_1=\ldots=x_n$. I was wondering if it was known whether a similar inequality holds for the exponential mean, written as $$EM_\alpha= \frac{1}{\alpha}\ln(\frac{1}{n}\sum_{i=1}^n e^{\alpha x_i})?$$

Answer 1

It suffices to substitute $e^x$ for $x$ and take the logarithm.