TITLE: Expected value of non-negative iid random variables
QUESTION [5 upvotes]: I want to show that, for $n$ iid non-negative random variables $x_1, \dots, x_n$, we have
$$
\mathbb{E} \max_{i \in [n]} x_i \lesssim (\mathbb{E} [x_i^{\log n}])^{\frac{1}{\log n }}.
$$
I can only go so far as to show $\mathbb{E} \max_{i \in [n]} x_i \leq n (\mathbb{E} [x_i^{\log n}])^{\frac{1}{\log n }}$ for $n \geq 3$, using Jensen's inequality and the inequality $\max_i x_i \leq \sum_i x_i$, which seems a bit wasteful (and responsible for the factor of $n$).
Any tips for getting rid of the factor of $n$ would be greatly appreciated!

REPLY [2 votes]: Here is another proof that the constant factor $e$ in mathworker21's nice proof is the best possible.
Suppose that $1-P(x_1=0)=P(x_1=1)=p:=p_n:=\dfrac{\ln n}n$. Then
$$E\max_{i\in[n]}x_i=1-(1-p)^n\ge1-e^{-np}=1-1/n\to1$$
and
$$(Ex_1^{\ln n})^{1/\ln n}=p^{1/\ln n}
=\exp\frac{\ln\ln n-\ln n}{\ln n}
\to e^{-1}.$$
So, the constant factor $e$ is the best possible.