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 $\|X\|_{\psi_2}$ and the sub-Gaussian parameter $\sigma_X$ for centered real-valued random variables. We show that $\sqrt{3/8} \cdot \|X\|_{\psi_2} \le \sigma_X \le \sqrt{\log 2} \cdot \|X\|_{\psi_2}$, and that both bounds are sharp, attained by the standard Gaussian and Rademacher distributions, respectively.