Consider the cone of continuously twice differentiable functions mapping positive reals to itself (i.e., $f\in C^2(\mathbb R_+)$ and $f\colon \mathbb R_+\to\mathbb R_+$) that satisfy

\begin{align} f(x)&\geq 0\\ f(x) - xf'(x)- x^2f''(x)&\geq 0 \end{align} for all $x\in\mathbb R_+$. Let us observe that this is indeed a convex cone in that scaling and adding two functions together would preserve these two inequalities.

My question is whether it would be possible/feasible to find the extreme rays (also called generators, I believe) of this convex cone.

One can observe that each power function $f(x) = x^p$ for $p\in[-1,1]$ is in this cone, as they are nonnegative and satisfy the differential condition. However it is not clear at all that these functions span this convex cone. Should one expect to have an explicit set of extreme rays?

Consider the cone of continuously twice differentiable functions mapping positive reals to itself (i.e., $f\in C^2(\mathbb R_{++})$ and $f\colon \mathbb R_{++}\to\mathbb R_{++}$) that satisfy

\begin{align} f(x)&\geq 0\\ f(x) - xf'(x)- x^2f''(x)&\geq 0 \end{align} for all $x\in\mathbb R_{++}$. Let us observe that this is indeed a convex cone in that scaling and adding two functions together would preserve these two inequalities.

My question is whether it would be possible/feasible to find the extreme rays (also called generators, I believe) of this convex cone.

One can observe that each power function $f(x) = x^p$ for $p\in[-1,1]$ is in this cone, as they are positive valued and satisfy the differential condition. However it is not clear at all that these functions span this convex cone. Should one expect to have an explicit set of extreme rays?