Suppose that $\phi: \mathbb D \to \Omega $ is a conformal map from the unit circle to a Jordan domain $\Omega$. Fix a point $e ^{i \theta} \in \mathbb D $, how I can show that function $u: \mathbb D \to \mathbb D$, which $u(z)= \arg ( \frac{\phi (z)-\phi(e^{i \theta})}{z-e^{i \theta}} ) $ is harmonic?
I can not prove this directly, for example by showing that $\triangle u =0$.
The fact that $u$ is harmonic is used in proof of Theorem II.4.2 of Garnett and Marshall's Harmonic measure.
The function $(\phi(z)-\phi(e^{i\theta}))/(z-e^{i\theta})$ is holomorphic and non-vanishing in $\Bbb D$: since the disk is simply connected it has a holomorphic logarithm. $\arg(w)$ is the imaginary part of $\log(w)$.
(If $w=re^{it}$ then $log(w)=\log(r)+it$.)