Problem 1793
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 \(\displaystyle r<|z|<\frac1r\). Difficulty: 5. |
Problem 1796
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 \(\displaystyle \overline{D}=\{\overline{z}:~z\in D\}\). Difficulty: 5. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |