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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?

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Sachchidanand Prasad
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  • Is this not a duplicate of the question you linked to? Zarrax's answer suggests why $f$ is identically $1$ (using the identity theorem @positrón0802 mentioned). – Jonas Meyer Jan 25 '17 at 06:55
  • That I have mentioned in my question that in that question he asked that which is correct, but I am asking which is incorrect. – Sachchidanand Prasad Jan 25 '17 at 12:26
  • If you know which are correct, you know which are incorrect. Zarrax's answer leads you to confirm your belief, which then makes answering the 4 specific questions easy. – Jonas Meyer Jan 25 '17 at 16:44

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