# List of risk-averse utility functions

In the context of optimal portfolio allocation, I am looking for a (possibly exhaustive) list of risk-averse utility functions verifying part of the so-called Inada conditions.

Essentially, I am looking for functions $$U : \mathcal{D}=[0,\infty) \to \Bbb{R}$$ verifying $$\frac{\partial U}{\partial x}(x) > 0\quad, \quad \forall x \in \mathcal{D}$$ $$\frac{\partial^2 U}{\partial x^2}(x) < 0\quad,\quad \forall x \in \mathcal{D}$$ $$\quad \lim_{x \to 0^+} \frac{\partial U}{\partial x}(x) \,\,\,= +\infty$$ $$\lim_{x \to +\infty} \frac{\partial U}{\partial x}(x) = 0$$

I could easily come up with the famous family of isoelastic utility functions (CRRA) among which power utility and the logarithmic utility.

However I was wondering whether there existed any other well known instances of such functions. I would be glad if someone could point me towards an answer or a rigorous approach for finding them.