Let $f(z)$ be analytic on $\mathbb{D}$ = {${z\in\mathbb{C}:|z-1|<1}$} such that $f(1) = 1$, if $f(z) = f(z^2)$ for all $z\in\mathbb{D}$, then which of the following statements is NOT correct?
1) $f(z) = [f(z)]^2$
2) $f(\frac{z}{2}) = \frac{1}{2}f(z)$
3) $f(z^3) = [f(z)]^3$
4) $f'(1) = 0$
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If we will assume $f(z)=1$ for all $z\in\mathbb{D}$ then this function clearly satisfies all the hypothesis so (2) is incorrect. I strongly believe that the only function which satisfy the above hypothesis is $f(z)=1$. But how will I prove this?