Problem 1796 (difficulty: 5/10)

Let \(\displaystyle f\) be holomorphic and non-vanishong on a convex domain \(\displaystyle D\). Assume that the boundary of \(\displaystyle D\) contains the real interval \(\displaystyle I\) and that \(\displaystyle f\) has a continuous extension to the interior of \(\displaystyle I\) where it satisfies \(\displaystyle |f|=1\). Show that \(\displaystyle f\) can be analytically continued to \(\displaystyle \overline{D}=\{\overline{z}:~z\in D\}\).

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