Averages of Nevanlinna counting functions of holomorphic self-maps of the unit disk
Let \(D\) denote the unit disc in the complex plane, let \(\phi\) be a holomorphic function mapping the disc into itself, and let \[ N_{\phi}(w) = \sum_{\phi(z) = w} \;\log{\frac {1} {| z| }} \] be the Nevanlinna counting function for the function \(\phi\) and the value \(w \in D\). The author gives an integral representation for \(N_{\phi}(a)\) of the form \[ N_{\phi}(a) = - \log{| a - \phi(0)| } + \int_{\partial D} \;\log{| a - \phi^{*}(\zeta)| } d\sigma(\zeta) - \mu_{a}(\partial D) \] where \(a \in D \setminus \{\phi(0)\}, \sigma\) is a Lebesgue linear measure, and \(\mu_{a}\) is the singular measure associated with the singular factor of the function \((a - \phi)/(1 - \bar{a} \phi)\). From this, some averaging formulas are developed for the counting function \(N_{\phi}(a)\) as \(a\) ranges over circles and discs centered at the origin, such as \[ \int_{\partial D} \;N_{\phi}(\rho \nu) d\sigma(\nu) = - \log^{+} {\frac {| \phi(0)| } {\rho}} - \int_{\rho}^{1} \log{\frac {t} {\rho}} d\sigma(\{\phi^{*}| > t\}) \] where \(\phi^{*}(\lambda)\) is the radial limit of \(\phi\) at \(\lambda \in \partial D\), and \[ \begin{multlined}\frac {1} {R^{2}} \int\int_{| a| < R} \;N_{\phi}(a) dA(a) = \log{\frac {1} {| \phi(0)| }} + \int_{\partial D} \;\log{| \phi^{*}(\zeta)| } d\sigma(\zeta)\\ + \frac {1} {2} \int_{| \phi^{*}| < R} \;\bigg\{ \bigg(\frac {| \phi^{*}(\zeta)| } {R}\bigg)^{2} - 1 - \log{\bigg(\frac {| \phi^{*}(\zeta)| } {R} \bigg)^{2}} \bigg\} d\sigma(\zeta)\end{multlined} \] for \(0 < R < | \phi(0)| \), where \(dA = 2 \rho \;d\rho \;d\sigma(\nu)\). As an application, a new proof is given for a result of \textit{T. Nakazi} [Proc. Am. Math. Soc. 131, 1267--1271 (2002; Zbl 1010.30022)] dealing with Rudin's orthogonal functions.
scientific article; zbMATH DE number 2140990