Computational Methods and Function Theory 9 (2009), No. 1, 161--184
Copyright Heldermann Verlag 2009
Janis Meyer
pmzjm2@maths.nott.ac.uk , University of Nottingham, School of Mathematical Sciences, Nottingham, NG7 2RD, U.K.
Let $\mu$ be a locally finite complex valued measure on $\mathbb{C}$ with an angular density with respect to a proximate order $\rho(r)$ and let $$ f_\mu(z)=z^p\iint_\mathbb{C} \frac{d\mu(\zeta)}{\zeta^p(z-\zeta)} $$ be the canonical Cauchy potential of $\mu$. The main part of this paper is devoted to an estimate of $f_\mu(re^{i\phi})$ uniformly in $\phi$ generalising some results of A.\ Gol'dberg, N.\ Korenkov and N.\ V.\ Zabolotskii on the asymptotic expansion of the logarithmic derivative of entire functions. As application we give a partial answer to a problem raised by A.\ Eremenko, J.\ Langley and J.\ Rossi concerning the evaluation of the Nevanlinna deficiency of zeros of functions of the form $$ f(z)=\sum_{k=1}^\infty \frac{a_k}{z-z_k}, \qquad \sum_{|z_k|\neq0}^\infty \left|\frac{a_k}{z_k}\right|<\infty $$ in terms of $$ \limsup_{r\to\infty} \frac{\log^+\left|\sum_{|z_k|\leq r}a_k\right|}{\log r}=\rho. $$ We discuss the result from the point of view of a theorem of Keldysh.
Keywords: Cauchy potentials, angular densities of measures, regularly distributed sequences, Nevanlinna theory.
MSC 2000: 30D30, 30D35, 31A15.
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