I'm trying to do a question but I have a doubt on holomorphic functions, here is the problem.
Let $A = \{z ∈ \mathbb{C} : \frac{1}{R}< |z| < R\}$. Suppose that $f : A → \mathbb{C}$ is holomorphic and that $|f(z)| = 1 $ if $|z| = 1$. Show that $f(z) = {\left(\overline{f(\bar{z}^{−1})}\right)}^{-1}$ when $ |z| = 1 $ and deduce that this holds for all $z ∈ A$.
Now, my problem is with the final deduction...
I think I need something about the zeroes of f as the right-hand side of the equality could not be defined at some point. Am I right or is there a way to prove that a function with these properties can never be zero in A?
Thank you all.