Abstract. A function $\varphi $ is called T-universal on an open set ${\cal O} \subset{\msbm C}$ if $\varphi $ is holomorphic on ${\cal O}$ and satisfies the following properties. For all compact sets $K$ with connected complement, for all functions $f$ which are continuous on $K$ and holomorphic in its interior and for all $\zeta\in \partial{\cal O}$ there exists a sequence $\{(a_n, b_n)\} $ in ${\msbm C}^2$ with $a_n z + b_n \in{\cal O}$ for all $z \in K$ and all $n \in{\msbm N}$, such that $\{a_n z + b_n\} $ converges to $\zeta $ and $\{\varphi(a_n z + b_n)\} $ converges to $f (z)$ uniformly on $K$. The existence of T-universal functions is proved for open sets ${\cal O}$ with simply connected components. If ${\cal O}$ contains the origin, then $\varphi $ can be chosen with a lacunary power series $\varphi(z) = \sum ^\infty_{\nu = 0} \varphi_\nu z^\nu $, where $\varphi_\nu = 0$ for $\nu\not\in Q$ with a certain prescribed set $Q \subset{\msbm N}_0$.

AMS Subject Classification (1991): 30B10, 30E10, 30B60

Received April 23, 1997 and in revised form September 4, 1997. (Registered under 2646/2009.)