Smooth universal Taylor Series

Let Omega subset of C be a simply connected domain in C, such that {infinity}U[C\(Omega) over bar is connected. If g is holomorphic in ohm and every derivative of g extends continuously on (Omega) over bar , then we write g is an element of A(infinity) (Omega).For g is an element of A infinity(Omega) and zeta is an element of (Omega) over bar we denote S-N(g,zeta) (z) = Sigma(N)(l=0) g(l) (zeta)/ll (z-zeta)(l).We prove the existence of a function f is an element of A infinity(Omega), such that the following hold: i) There exists a strictly increasing sequence mu n is an element of {0,1,2...},n = 1,2,.....such that, for every pair of compact sets Gamma,Delta subset of (Omega) over bar and every l is an element of {0,1,2,....} we have (zeta is an element of Gamma w is an element of Delta)sup sup partial derivative l/partial derivative wl S mu n(f,zeta)(w)-f((l))(w)-> 0, as ->+infinity and ii) For every compact set K subset of C with K boolean AND ($) over bar = 0 and K-c connected and every function h : K -> C continuous on K and holomorphic in K-0, there exists a subsequence {mu(l)(n)}(n=1)(infinity),such that, for every compact set L subset of (Omega) over bar we have (zeta is an element of L z is an element of K)sup sup S mu ln(f,zeta)(z)-h(z)-> 0, as n ->+infinity.