Suppose that $X_1, \ldots, X_n \sim N(0,1)$ are independent random variables. I am interested in finding a constant C that satisfies:
$$ E\left[\max_{1\leq i\leq n}|X_i|\right] \leq C \sqrt{log\ n} $$
I know one method is to employ the moment generating function trick, then take logs of both sides. However, I was wondering if there exists a more direct method. thanks!