Composition and differentiation operators and fast approximation

Let C=(Cn)n∈NC=(Cn)n∈N and D=(Dn)n∈ND=(Dn)n∈N be families of composition and differentiation operators, respectively, i.e., Cnf=f∘φn,Df=f′, where ff is holomorphic on some domain Ω⊆CΩ⊆C. Our main question is: How fast can a totally bounded set MM of holomorphic functions, in other words a normal family, be approximated by the “orbit” {Cnf:n∈N}{Cnf:n∈N} or {Dnf:n∈N}{Dnf:n∈N}, respectively, of one suitably constructed function ff? Our answer consists of upper bounds for the numbers F(f,1/n):=inf{N∈N:Any g∈M is approximable with error <1/nby 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 DD, or the set S:={f biholomorphic on D:f(0)=0,f′(0)=1}.S:={f biholomorphic on D:f(0)=0,f′(0)=1}.