First question: Let f(z) be an analytic function defined in a disk |z| < r. Show that the functional equation f(2z)=f(z)f '(z) satisfied on |z| < r/2 implies that f(z) is an entire function.
Second Question: Let f(z) be a power series in z satisfying the equation f(z)=z-f(z^2). Compute the convergence radius of f(z).
Would I be correct in saying that for the second question the radius of convergence is 1? By iterating f(z)=z-f(z^2), we get f(z)=z-z^2+z^4-z^8+... which converges for |z|<1.