TITLE: Is the Binomial Expectation of Convex Function Convex in p?
QUESTION [6 upvotes]: Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function.
Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex in $p$?
Related questions:
Binomial Expectation of Convex Function and expected values over binomial distributions
Note: Based on trial-and-error analysis, it appears to be convex. If anyone is interested, I can send them a spreadsheet with some numbers.

REPLY [3 votes]: This is known, for more general exponential families in place of the binomial one,
from papers of Shaked (1980) and Schweder (1982), see http://www.jstor.org/stable/10.2307/2984960 and http://www.jstor.org/stable/10.2307/4615873.