A generic result concerning univalent universal functions

Abstract.

Let $\Omega$ be a Jordan region. We prove that generically every function $f \in H(\Omega)$ , univalent in $\Omega$ and continuous on $\overline\Omega$ is universal under derivatives and universal in respect to overconvergence. In fact f realizes both approximations with the same approximative sequence. The proof is based on a modified version of the classical result of J. Walsh concerning approximation of holomorphic functions by polynomials.