Let f : D C be a holomorphic function such that f(z2) = (f(z))2 for all z D, and D = D_1 (0) is the unit disk. (a) Show that f(0) is 0 or 1. (b) Show that f(z^{2n}) = (f(z))^{2n} for all z D and all n N. (c) Suppose f(0) = 1. Use the equation f((z_0)^2n) = (f(z_0))^2n for a point z_0 D \ {0} and the previous question to show that f is constant equal to 1. (d) Now suppose that f(0) = 0 but that f is not constant. Let m 1 be the order of zero from f to 0 and let g : D C be a holomorphic function such that f(z) = z^m*g(z) for all z D. Show that g also satisfies the equation g(z^2) = (g(z))^2, then deduce that f(z) = z^m for all z D.