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If $f:\overline{\mathbb{D}}\longrightarrow\mathbb{C}$ is a holomorphic function over $\mathbb{D}$ and $f(\mathbb{T})\subset \mathbb{R}$ then is $f$ constant?

Consider:

$f:\overline{\mathbb{D}}\longrightarrow\mathbb{C}$ continuous

$\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$

$\overline{\mathbb{D}}=\{z\in\mathbb{C}:|z|\le1\}$

$\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$

Any hint would be appreciated.

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The imaginary part of a holomorphic function is harmonic, and it attains its maximum on the border of any compact set contained in $\mathbb{D}$, taking the limit on all such compact sets we get that that maximum is $0$, that is $\Im{f}=0$. You now have an holomorphic function with zero imaginary part, so by the Cauchy Riemann equation you conclude.

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I don't know if this will work, but this is what I think when I see the problem given.

Complex analysis gives us a number of tools for reconstructing a function given its values on various sets; in this case, we have a direct formula (Cauchy's Integral formula, I think the name is) for obtaining the value of a function inside of a loop from its values on the loop.

So I would start by looking at writing down said formula and seeing if it implies anything.


My other thought from the problem is that $\log(z)$ is aligned with these sorts of domains. I would spend some time experimenting with functions related to $\log(z)$ to see if I could construct one that is holomorphic in the disk and real on the boundary.

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