Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as
$$ E[F(X)] = F(a) + E[ O((X-a))] $$
Under what conditions can I write $$ E[F(X)] = F(a) + O(E[(X-a)]) \,\,? $$
And if there is no condition under which the two are equivalent, any ideas how I could quantify
$$ E[ O((X-a))] - O(E[(X-a)]) \,\, ? $$
I suspect convexity will play a role, but not clear yet to me how.