Let $f(z)$ be analytic function on $D = \{z\in C : |z-1|<1\}$ such that $f(1) = 1$. If $f(z) = f(z^2)$ for all $z\in D$. Then which of the following statement is not correct.
1-$f(z) = [f(z)]^2$ for all $z\in D$
2- $f(z/2)$ = $\frac{1}{2}[f(z)]$ for all $z\in D$
3- $f(z) = [f(z)]^3$ for all $z\in D$
4- $f'(1) = 0$
I am fully stucked on this problem. I need help. Thanks