Orlicz norm of random variable and variance
The Kearns-Saul inequality states that if $X\sim Ber(p)$ then
$$ E[\exp(t(X-p))] = (1-p)e^{-tp}+pe^{t(1-p)} \le \exp\left(\frac{1-2p}{4\log((1-p)/p)}t^2\right).$$

The subgaussian constant $\frac{1-2p}{4\log((1-p)/p)}$ is optimal.
See http://ecp.ejpecp.org/article/view/2359 and especially the appendix in http://www.jmlr.org/papers/v16/berend15a.html for background and a slick proof.

The K-S inequality is a considerable improvement over Hoeffding and Bernstein for very small/large $p$.