This is an exercise in Conway. Here $G$ is an open set of the complex plane, C(G,{C}) is the set of all continuous functions from $G$ to the complex plane and $H(G)$ is the set of all analytic functions on $G$. I managed to show (b), but am stuck at (c). I think I have to use the definition of derivatives of the analytic functions but can't proceed on. Could anyone please help me?
If $D = D_r(z_0)$ is a disk with $\overline D \subset G$ then for $a, b \in D$ Cauchy's integral formula gives $$ f(a) - f(b) = \frac{a-b}{2 \pi i} \int_{\partial D} \frac{f(z)}{(z-a)(z-b)} dz \, . $$ It follows that if $\cal F$ is uniformly bounded on $D$ then it is uniformly Lipschitz on $D_{r/2}(z_0)$.