The distribution of values of an inner function (Q801185)

Let \(\phi\) be an inner function in the unit disc U. For \(\alpha\in U\), \(\phi_{\alpha}=(\phi -\alpha)/(1-{\bar \alpha}\phi)\). If n(r,\(\alpha)\) is the number of zeros of \(\phi_{\alpha}\) whose moduli are at most r and \(\delta\) (\(\alpha)\) is the total mass of the singular measure associated to the inner function \(\phi_{\alpha}\) then \[ L(r,\alpha)=- (2\pi)^{-1}\int^{2\pi}_{0}\log | \phi_{\alpha}(re^{i\theta})| d\theta \] satisfies \(L(r,\alpha)=\int^{1}_{r}n(t,\alpha)t^{-1}dt+\delta (\alpha)\) using Jensen's formula. Let \(\Delta (r)=(2\pi)^{-1}\int^{2\pi}_{0}(1- | \phi (re^{i\theta})|^ 2)d\theta.\) The author studies the ratio \(L(r,\alpha)/\Delta (r)\) and shows, among other things, that \(0<\lim \inf_{r\to 1}L(r,\alpha)/\Delta (r)\) for all \(\alpha\in U\), and \(\lim \inf_{r\to 1}L(r,\alpha)/\Delta (r)<\infty\) for all \(\alpha\in U\) with the exception of a set of capacity zero.