Analytic function with restricted range

Question

If $f (z)$ be analytic function on $D =\{z \in\Bbb C : |z-1| <1 \}$ such that $f (1) =1$ , if $f (z) =f (z^2)$ for each $z\in D$ , then which one of the following statement is not correct ?

(a) $f (z) =[f (z) ]^2$ for each $z\in D$

(b) $f (z/2) =f (z) /2$ for each $z \in D $

(c) $f (z^3) = [f (z)]^3$ for each $x \in D$

(d) $f '(1) =0$

Is $f (z)$ a constant function ? If yes , then what is the reason?

Answer 1

Hint : For (d)

$$f'(1)=\lim_{h \to 0}\frac{f(1+h)-f(1)}{h}=\lim_{h \to 0}\frac{f((1+h)^2)-1}{h}=\lim_{h \to 0}\frac{2(1+h)f'((1+h)^2)}{1}=2f'(1)$$