Higher order Jensen-like expansion upper bound

Question

If $Z$ is a random variable with fine moment generating function, what is a good way to upper bound $$|\log \mathbb{E}e^Z- \mathbb{E}Z- \frac{1}{2}\mathbb{E}Z^2|$$

This looks like a third offer Taylor series but cannot put finger on it.

Answer 1

If $M(t) = \mathbb E[e^{tZ}]$ is the moment generating function, $\mathbb E[e^Z] = M(1)$. $|Z| \le (1+Z^2)/2$ so $|Z + Z^2/2| \le 1/2 + Z^2 \le 2 \cosh(Z)$. Thus $$ \left| \log \mathbb E[e^Z] - \mathbb E[Z] - \frac{1}{2} \mathbb E [Z^2] \right| \le |\log M(1)| + M(1) + M(-1) $$