Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking a derivative with respect to the boundary point $b$.
$g$ is a positive, strictly increasing and convex function defined on the reals, and thus the expectation of $g$ is greater than $g$ of the expectation:
$$ \frac{\int_{a}^{b}g(t)\text{ }f(t)\text{ }dt}{\int_{a}^{b}f(t)\text{ }dt}% -g\left( \frac{\int_{a}^{b}t\text{ }f(t)\text{ }dt}{\int_{a}^{b}f(t)\text{ }% dt}\right) \geq 0 $$
(Jensen's difference)
Question: does this difference increase with respect to $b$ ?
Intuitively, as we expand the domain to the right we are including more of the even faster growing parts of $g$ so that the answers seems to be yes.
When I take the derivative of each term with respect to $b$, cancel and rearrange I find that the answer is "yes", if and only if $$\frac{g(b)-E\left[ g(t)\mid a\leq t\leq b\right] }{b-E\left[ t\mid a\leq t\leq b\right] }\geq g^{\prime }\left( E\left[ t\mid a\leq t\leq b\right] \right) $$
Looking at a proof of Jensen's inequality using a supporting line almost helps here, but not quite. This is where I am stuck.
Is the difference increasing? Or is there a condition on $f$ such that it is?