Integral formula involving conformal mapping

Question

Suppose $f$ is analytic in a domain $\Omega$ that contains the closed unit disk. Show that for every $z$ in the unit disk, $f(z)=\frac{1}{2\pi}\int_0^{2\pi} f(\frac{e^{i\theta}+z}{1+\bar{z}e^{i\theta}}) d\theta$.

I believe that we need to apply Cauchy's integral formula and the conformal self maps of the unit disk. But I have no idea how to put them together. Can someone help me? Thanks

Answer 1

Hint: For each $z\in \mathbb D,$

$$g_z(w) = \frac{z+w}{1+\bar z w}$$

is an automorphism of the unit disc. You are trying to prove

$$(f\circ g_z)(0) = \frac{1}{2\pi}\int_0^{2\pi} (f\circ g_z)(e^{it})\, dt.$$