The Chernoff exponent is a Legendre transform

Let $X_1,\dots,X_n$ be i.i.d. with cumulant generating function $\psi(\lambda) = \log\E e^{\lambda X_1}$ finite near $0$. Show that the Chernoff bound for $\bar X_n$ reads

$$\P(\bar X_n \ge a) \le e^{-n\psi^*(a)}, \qquad \psi^*(a) = \sup_{\lambda>0}\big(\lambda a - \psi(\lambda)\big),$$

and compute $\psi^*$ explicitly for (i) $X_1\sim\Ncal(0,\sigma^2)$ and (ii) $X_1$ Bernoulli$(p)$. Verify in each case that $\psi^*(a)$ is non-negative, zero exactly at $a = \E X_1$, and convex.