I met one technical problem in my project, and I wish some reference here..
Suppose $\{X_i\}_{i=1}^n$ are independent random variables. Are there some inequalities that give upper bound of $\mathbb{E}[\max_{1\leq i\leq n}X_i]$?
By now I only know one such inequality give by Aven (1985), that is, if $\{X_i\}_{i=1}^n$ is a sequence of random variables, then
$\mathbb{E}[\max_{1\leq i\leq n}X_i] \leq \max_{1\leq i\leq n}\mathbb{E}[X_i]+\sqrt{\frac{n-1}{n}\sum_{i=1}^nVar(X_i)}$
Above inequality does not use the fact that $\{X_i\}_{i=1}^n$ are independent. I was wondering if there are some other similar inequalities that I can consider.
Many thanks.