Consider a $C^{\infty}$ nonnegative function $f(x,y,z)$, $x,y,z>0$ and let $\lambda f(\lambda x, \lambda y,\lambda z) \equiv f(z,y,z)$ for any $\lambda > 0$ (positive homogenity). Define $$ g_{\alpha}(x,y) := f(x^{\alpha},y^{\alpha},z). $$ I'm interested in such numbers $\alpha>0$ that $g_{\alpha}(x,y)$ is a completely monotone function for any $z>0$, i.e. $$ (-1)^{m+n}\frac{\partial^{m+n} g_{\alpha}(x,y)}{\partial x^m \partial y^n} \geqslant 0. $$ I denote a set of such $\alpha$ by $A_{f}$. In other words, $A_{f}$ is a set of such $\alpha$ that $f(x^\alpha,y^\alpha,z)$ is a completely monotone function of $(x,y)$ for any $z$. For example, $A_{0} = (0,+\infty)$.
The easiest question that arises here and seems very nontrivial for me is if it is possible to find a function $f(x,y,z)\neq0$ such that $|A_{f}| \geqslant 2$? I'm not sure that one can find something in literature precisely on this topic but I think that somebody could encounter related things in its work. In this relation any help is very appreciated.
