To find analytic function with given condition How to find  all analytic function on the disc $\{z:|z-1|<1 \}$ with $f(1)=1$ and $f(z)=f(z^2)$ ?.

Since the domain $\left\{ | z| < 1\right\}$ is included in the domain of convergence of the logarithm, you can write $w = \log z$ and $g(w) = f(z) = f(\exp w)$. Your conditions then becomes $g(2w) = g(w)$. For $g$ continuous at zero, this is enough to show that $g$ is constant (because, for any $w$, $f(w) = f(w/2) = f(w/4) = \dots$, and this sequence has $f(0)$ as a limit because of continuity).
