T-universal Functions With Prescribed Approximation Curves

Let be F a family of curves in the unit disc. We show that the set of all functions f holomorphic on the unit disc, which satisfy the following condition, is G-delta and dense in the space of all functions holomorphic on the unit disc: For each compact set K with connected complement, each function g continuous on K and holomorphic on its interior, every point t on the unit circle, every curve C in F (ending in t) and any e>0 there exist numbers 0<a<1 and b in C such that |f(az+b)-g(z)|<e for all z in K and |b-t|<e. The set of these functions is called the class of T-universal functions with prescribed approximation curves.