I am trying to derive the following equation for an analytic function, $f$, defined on the unit disc, $D$
\begin{equation*} f(z)=\frac{1}{\pi}\int_{D} \frac{f(w)}{(1- \bar{w}z)^{2}}dA(w). \end{equation*}
Question: Anyone have any ideas on how to derive this formula?
I've been able to verify that the above is true for $z =0$. In attempting to derive for the general case, I used conformal maps to arrive at
\begin{equation*} f(z)=\frac{1}{\pi}\int_{D} f(w) \left| \frac{-1+|z|^{2}}{(-1+w\bar{z})^{2}} \right|^{2} dA(w) \end{equation*}
Unfortunately this is not the formula I was trying to derive,