Let $F:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $F(1)=0$, which is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$.
Suppose also that $F$ is convex on $(0,1]$.
Finally, let $g:X \to (0,\infty)$ be a measurable function defined on a probability space $(X,\Sigma,\mu)$, with $\int_X g \in (0,1)$.
Does $F(\int_X g) \le\int_X F\circ g\,\,$ ?
If $\text{Image}(g)$ was contained in $(0,1]$, this would follow from a standard application of Jensen inequality. Here I am allowing $g$ to obtain values which lie outside the domain where $F$ is convex.
Best Answer
Yes.
The point $\left(\int_X g, \int_X F\circ g\right)\in\mathbb R^2$ is contained in the convex hull of the set $G=\{(x,F(x)): x\ge0\}$, which in turn is contained in the epigraph $S$ of the convex function $F^*$ given by $F^*(x)=F(\min(x,1))$. Note that $F^*\le F$ pointwise, and $F(x)=F^*(x)$ for $x\in[0,1]$. Since $\int_X g\in(0,1)$, we have $F\left(\int_X g\right)=F^*\left(\int_X g\right).$ Since $F^*$ is convex, we have $F^*(\int_X g)\le \int_X F^*{\circ} g$ and then $F\left(\int_X g\right)\le\int_X F{\circ} g,$ as desired.
Added, in reaction to a comment:
Expressed in a different language, your problem is this: Suppose $F$ is as you specify, and the non-negative random variable $X$ satisfies $E[X]\in(0,1)$; you want to know if the expectation of the random vector $(X,Y)=(X,F(X))$ lies above the graph of $F$. In general vector expectations are elements of the closed convex hull of their vectorial support. In your case the vectorial support is the set I call $G$ above. Your probability measure $\mu$ induces a distribution of mass on $G$; the centroid of this distribution is the vectorial expectation $(E[X],E[Y])$. The convex hull of $G$ is of course convex, without any assumption on $F$. In your special case, $F$ restricted to $[0,1]$ is convex, and $F^*$ defined by $F^*(x)=F(x)$ for $x\in[0,1]$ and $F^*(x)=0$ for $x>1$ is also convex. So Jensen applies to $F^*$ and so on.
I thought of this method in the current problem because I think of all Jensen inequality arguments this way, in "moment space". There is an underlying probability measure, with a centroid, and what is desired is to know if the centroid is contained in a certain convex set. In your case, the epigraph of $F$ need not be a convex set, but its intersection with the half plane cut out by $x\le 1$ is convex.