In an economic model I am working with I would like to establish that the difference of two special convex functions is convex. I know that in general the difference of convex functions need not be convex, see here, but I was hoping someone can point out if the properties of my functions suffice or can provide a counterexample that might suggests what additional property is needed.
Suppose $s_0,\ s_1: [0, 1] \rightarrow \mathbb{R} \bigcup \{+\infty, -\infty\}$ are convex, is $s_1 - s_0$ convex in $[0,1]$ under the following assumptions?
Assume that $s'_1(0) > s'_0(0)$ and that there are two strictly increasing and continuous functions $g_0,\ g_1: [0, 1] \rightarrow [0, 1]$, with $g_0(0) = g_1(0) = 0$, $g_0(1) = g_1(1) = 1$, $g_1(t) \geq g_0(t)$ for all $t \in [0,1]$, and such that $$s_i(t) = \frac{1}{t} \int^t_0 \frac{g_i(s)}{1-g_i(s)} ds \quad \text{for } i = 0, 1$$
Note that under these assumptions $s_1(t) \geq s_0(t)$ for all $t \in [0,1]$, as in the post mentioned above; in addition, $s_0(0) = s_1(0) = 0$ and $s'_0(1) = s'_1(1) = \infty$, in contrast to the counterexamples mentioned in the post mentioned above.
