Composition and differentiation operators and fast approximation

Let C = (Cn)n∈N and D = (D)n∈N be families of composition and differentiation operators, respectively, i.e., Cnf = f ◦ φn, Df = f ′, where f is holomorphic on some domain Ω ⊆ C. Our main question is: How fast can a totally bounded set M of holomorphic functions, in other words a normal family, be approximated by the “orbit” {Cnf : n ∈ N} or {Df : n ∈ N} respectively, of one suitably constructed function f? Our answer consists of upper bounds for the numbers F (f, 1/n) := inf{N ∈ N : Any g ∈M is approximable with error < 1/n by the first N elements of the orbit of f}, n ∈ N. In particular, we calculate such bounds for well-known classical normal families, like the biholomorphisms of the unit disk D, or the set S := {f biholomorphic on D : f(0) = 0, f ′(0) = 1}.