Analytic functions $g$ on $|z|<1$, with $g'(1/n)=2g(1/n)$ for $n=2,3,\dots$

Question

Analytic functions $g$ on $|z|<1$, with $g'(1/n)=2g(1/n)$ for $n=2,3,\dots$

How to determine all such functions? I am thinking using Schwarz–Pick theorem but so far haven't figured out. Any ideas?

Answer 1

Hint: You have $g'=2g$ on a set with limit point in the domain. Apply the identity principle. Then consider $e^{-2z}g(z).$