a. Let \(\Omega \) be an open set containing the closed unit disk, \(f\) holomorphic on \(\Omega \), and suppose that \(f\) has an analytic logarithm on \(\Omega \). Show that if \(z=re^{i\theta }\) with \(0\le r<1\) then \[ \ln |f(z)|=\frac {1}{2\pi }\int _0^{2\pi }P_r(\theta -t)\ln |f(e^{it})|\,dt; \] \(P_r(\theta )\) denoting the Poisson kernel.
b. Let \(g\) be holomorphic on \(\Omega \) and assume that \(g(z)\ne 0\) for all \(|z|=1\). If \(a_1,\ldots ,a_n\) are the zeros of \(g\) in the open unit disk, repeated according to their multiplicities, \(z=re^{i\theta }\), \(0\le r<1\), \(g(z)\ne 0\), then show that \[ \ln |g(z)|=\sum _{j=1}^{n}\ln \left |\frac {z-a_j}{1-\bar a_j z}\right | +\frac {1}{2\pi }\int _0^{2\pi }P_r(\theta -t)\ln |g(e^{it})|\,dt. \]