Sharp constants relating the sub-Gaussian norm and the sub-Gaussian parameter

We determine the optimal constants in the classical inequalities relating the sub-Gaussian norm $\mathrm{\parallel }X{\mathrm{\parallel }}_{{\psi }_2}$ and the sub-Gaussian parameter ${\sigma }_X$ for centered real-valued random variables. We show that $\sqrt{3\mathrm{/}8}\cdot \mathrm{\parallel }X{\mathrm{\parallel }}_{{\psi }_2}\le {\sigma }_X\le \sqrt{\log 2}\cdot \mathrm{\parallel }X{\mathrm{\parallel }}_{{\psi }_2}$, and that both bounds are sharp, attained by the standard Gaussian and Rademacher distributions, respectively.