If $\phi:\mathbb{S}^1\to \mathbb{S}^1$ is a homeomorphism then its harmonic extension on $\mathbb{D}$ is given by, $$ f(z)=\dfrac{1}{2\pi}\int_0^{2\pi} \dfrac{1-|z|^2}{|e^{it}-z|^2}\phi(e^{it}) .$$
I was able to prove the following things:
- $f$ is harmonic on $\mathbb{D}$ and is continuous on $\overline{\mathbb{D}}.$
- $f$ is locally univalent (one-to-one)
I tried to show the surjectivity, by saying that if it is not surjective, then let $f$ misses at least one point, say $w$ from co-domain $\mathbb{D}$. Now since $f$ is continuous and hence will miss a ball around $w$ (the main thing is that the ball may be open entirely in the interior or may be open in $\overline{\mathbb{D}}$. After that, I am unable to get a contradiction.
Any help will be appreciated. Thanks in advance.