Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector.

Question

$\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$

Observation

This paper allows us to upper-bound things like $\Delta_f(z,p):=f(\sum_{i}z_i p_i) - \sum_i f(p_iz_i)$, thus providing a kind of reversed Jensen's inequality.

Indeed, it was shown that

If $f$ is concave, $a:=\min_i z_i$ and $b := \max_i z_i$, then $$ \Delta_f(z,p) \le \max_{p,q \ge 0,\;p+q=1} pf(b)+qf(a) - f(pb+qa). $$

I could probably use this with $f=\log$ to bound $\Delta_{\log} (z,p)$.