Let $\phi_{z}(w)=\frac{z-w}{1-w\bar{z}}$ which is a conformal mapping of $\mathbb{D}$. $f\in C(\bar{\mathbb{D}})\cap C^2(\mathbb{D})$. $\mathbb{D}$ is the unit disk centered at the origin in the complex plane. $dA(w)$ is the normalized lebesgue measure on $\mathbb{D}$.
Define $G[f](z)=\int_{\mathbb{D}}(\log|w|^2)f({\phi}_z(w))\frac{(1-|z|^2)^2}{|1-w\bar{z}|^4}\,dA(w)$ , how to see that $G[f]\in C^2(\mathbb{D})$?
I encounter this statement in a proof of a theorem related to Green function. The author says it is clearly to get the twice continuity. But i really can not see why. Please help.