On a class of meromorphic functions with three singular values

We give a parametrization of the class of real meromorphic functions in the plane for which the preimage of $\{0,1,\infty\}$ is contained in the real line, and all singular values belong to $\{0,1,\infty\}$. As corollaries, we obtain a new parabolic type criterion and prove that all such functions are limits of rational functions of the same class.