Prove that $E[\max_{1\leq i\leq n}\vert X_i\vert]\leq C\sqrt{\log(2n)}\sigma$ for some constant $C \in R$, where $X_1, ..., X_n\sim \mathcal{N}(0,\sigma)$ are i.i.d.
Hint: Use Jensen's inequality: $f(E[Y])\leq E[f(Y)]$ for any convex function $f$.
I think that we can make use of the moment-generating function. $$E[e^{\lambda X}]=e^{\sigma^2\lambda^2/2}$$ where $\lambda\in R$ is a constant, and $X\sim\mathcal{N}(0, \sigma)$.
Moreover, i have tried to use $\max_{1\leq i\leq n}\vert X_i\vert\leq \sum_{i}^n \vert X_i\vert$, yet no good ideas.
Assume $\lambda>0.$
$$ \begin{aligned} \mathbb{E}[\max_{1\leq i\leq N}X_i]&=\mathbb{E}[\frac{1}{\lambda}\log e^{\lambda\max_{1\leq i\leq N}X_i}]\\ &\leq \frac{1}{\lambda}\log\mathbb{E}[e^{\lambda\max_{1\leq i\leq N}X_i}]\\ &=\frac{1}{\lambda}\log\mathbb{E}[\max_{1\leq i\leq N}e^{\lambda X_i}]\\ &\leq \frac{1}{\lambda}\log\sum_{i=1}^N\mathbb{E}[e^{\lambda X_i}]\\ &\leq \frac{1}{\lambda}\log \sum_{i=1}^Ne^{\lambda^2\sigma^2/2}\\ &=\frac{\log n}{\lambda}+\frac{\lambda\sigma^2}{2} \end{aligned} $$
Let $\lambda=\sqrt{2(\log N)/\sigma^2}$, we can get $\mathbb{E}[\max_{1\leq i\leq N}X_i]\leq \sqrt{2\sigma^2\log n}$
Moreover, there is a fact that $\max_{1\leq i\leq N}\vert X_i\vert=\max_{1\leq i\leq 2N}X_i$, where $X_{N+i}=-X_i\sim\mathcal{N}(0,\sigma^2)$
Thus $\mathbb{E}[\max_{1\leq i\leq N}\vert X_i\vert]\leq \sqrt{2\log (2N)}\sigma=C\sqrt{\log(2 N)}\sigma$