  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.
