Problem 1793 (difficulty: 5/10)

Let \(\displaystyle f\) be a holomorphic function on \(\displaystyle r<|z|<1\) which extend continuously to the unit circle and satisfies (a) \(\displaystyle f(z)\in \R\) for \(\displaystyle |z|=1\) (b) \(\displaystyle f\ne0\), and \(\displaystyle |f(z)|=1\) for \(\displaystyle |z|=1\). Prove that \(\displaystyle f\) has an analytic continuation to \(\displaystyle r<|z|<\frac1r\).

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