An $L^2$ mean inequality and normality of an $L^2$-bounded holomorphic family

Show that if \(f\) is analytic in an open set containing the disk \(|z-a|\le R\), then \[ |f(a)|^2\le \frac{1}{\pi R^2}\int_0^{2\pi}\int_0^R |f(a+re^{i\theta})|^2 r\,dr\,d\theta. \]

Let \(\Omega\) be a region and \(M>0\). Let \(\mathcal F\) be the family of all analytic functions \(f\) on \(\Omega\) such that \[ \iint_\Omega |f(z)|^2\,dx\,dy\le M. \] Show that \(\mathcal F\) is a normal family.