Cauchy potentials with angular density measures and a generalisation of a theorem of Keldysh (Q2378575)

Let \(\mu\) be a locally finite complex valued measure on \(\mathbb C\). Denote by \(\|\mu\|:=\mu_{+,1}+\mu_{-,1}+\mu_{+,2}+\mu_{-,2}\) the positive measure obtained from the Jordan decomposition \(\Re \mu=\mu_{+,1}-\mu_{-,1}\), \(\Im \mu=\mu_{+,2}-\mu_{-,2}\). For a proximate order \(\rho(r)\) consider the angular density \(\Delta(\psi):=\lim_{t\to\infty} t^{-\rho(t)}n(t,\psi)\) where \(n(t,\psi):=\mu(\{re^{\psi}: r\leq t, -\pi\leq \psi\leq \psi\})\). It is assumed that the angular density exists for all but countable many \(\psi\in]-\pi,\pi]\) and there exists \(\|\Delta\|:= \lim_{r\to\infty }r^{-\rho(r)}\|n(r,\pi)\|\). At first the author obtains formulae for the asymptotical behavior of the Cauchy potential \[ f_\mu(z):=z^p\int\!\!\!\int_{\mathbb C}{d \mu(\zeta)\over \zeta^p(z-\zeta)}. \] The results generalize some theorems by \textit{A.~Gol'dberg, N.~Korenkov} and \textit{N. Zabolotzkii} on the asymptotical behavior of the logarithmic derivative of entire functions. Then, as application, a partial answer is given to a problem by \textit{A.~Eremenko, J.~Langley} and \textit{J.~Rossi} [J. Anal. Math. 62, 271--286 (1994; Zbl 0818.30020)] on the Nevanlinna deficiency of zeroes of functions of the form \(f(z)=\sum_{k=1}^\infty a_k/(z-z_k)\) with \(\sum_{z_k\not=0}\left|a_k/z_k\right|<+\infty\). The results generalize a theorem by Keldysh.