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- Jan 26, 2012

- 995

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**Problem**: Suppose $f$ is a non-vanishing continuous function on $\overline{\mathbb{D}}$ that is holomorphic in $\mathbb{D}$, where $\mathbb{D}=\{z\in\mathbb{C}:|z|\leq 1\}$ is the unit disc. Prove that if$$|f(z)|=1\quad\text{whenever }|z|=1,$$

then $f$ is constant.

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**Hint**:

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