Carefully state and prove Jensen's inequality for a convex function $c: I \rightarrow \mathbb{R}$, where $I \subseteq \mathbb{R}$ is an interval. Assuming that $c$ is strictly convex, give necessary and sufficient conditions for the inequality to be strict.

Let $\mu$ be a Borel probability measure on $\mathbb{R}$, and suppose $\mu$ has a strictly positive probability density function $f_{0}$ with respect to Lebesgue measure. Let $\mathcal{P}$ be the family of all strictly positive probability density functions $f$ on $\mathbb{R}$ with respect to Lebesgue measure such that $\log \left(f / f_{0}\right) \in L^{1}(\mu)$. Let $X$ be a random variable with distribution $\mu$. Prove that the mapping

$$f \mapsto \mathbb{E}\left[\log \frac{f}{f_{0}}(X)\right]$$

has a unique maximiser over $\mathcal{P}$, attained when $f=f_{0}$ almost everywhere.