While reading Garnett's book about bounded analytic functions I faced with the following integral $$F(z)=\int\limits_{|\zeta|<1} \frac{G(\zeta)}{z-\zeta}d\zeta \wedge d\bar{\zeta},$$ where $G$ is $C^1$ function in the unit disc. It is claimed that F is $C^2$ in the unit disc, but I cannot understand why. Can somebody give a solution or a hint?
Thanks for any help in advance.