Given $n$ independently distributed sub-gaussian random variable $Z_i$, each has bounded MGF due to sub gaussian property:

\begin{eqnarray} && E[e^{Z_i}] \leq e^{\lambda^2 \sigma^2 /2 } \;\;\; \forall i = 1 \dots n \end{eqnarray}

for some $\lambda$ being postive real number. Let $X = \mathop{\max}_{1\leq i \leq n } Z_i$, one has that:

\begin{eqnarray} && E[X] \leq \sigma \sqrt{2 \log n} \end{eqnarray}

To prove this, note that the aim is to upper bound the expected value $E[X]$; thus Jensen’s inequality comes in handy:

\begin{eqnarray} && e^{t E[X]} \leq E[e^{t X}] = E[\mathop{\max}_{1\leq i \leq n} e^{t Z_i}] \leq \sum _{i=1} ^{n} E[e^{t Z_i}] \leq n e^{t^2 \sigma^2 /2} \end{eqnarray}

Thus \begin{eqnarray} && E[X] \leq \frac{\log n}{t} + t \sigma ^2 /2 \leq \sigma \sqrt{2 \log n} \end{eqnarray}