Let $f,g$ be holomorphic function in $mathbb{D}$ that are continuous in $overline{mathbb{D}}$. Show that if $f=g$ on $|z|=1$, then $f=g$

Mathematics Asked on January 5, 2022

Let $$f,g$$ be holomorphic function in $$mathbb{D}$$ that are continuous in $$overline{mathbb{D}}$$. Show that if $$f=g$$ on $$|z|=1$$, then $$f=g$$

It seems like identity theorem. But they have to be equal on an open connected set. $$overline{mathbb{D}}$$ is not open so I cannot use the identity theorem. At least directly.

Let $$varphi=f-g$$, then $$varphi$$ is holomorphic on $$mathbb{D}$$ and thus $$forall zinmathbb{D},|varphi(z)|leqslantmax_{|u|=1}|varphi(u)|=0$$ and thus $$varphi=0$$ on $$mathbb{D}$$ and $$f=g$$ on $$overline{mathbb{D}}$$.

Answered by Tuvasbien on January 5, 2022

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