# If $f[\mathbb{T}]\subset \mathbb{R}$ then $f$ is constant

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.

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.
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.