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 Ufimsk. Mat. Zh., 2012, Volume 4, Issue 1, Pages 122–135 (Mi ufa138)

The angular distribution of zeros of random analytic functions

M. P. Maholaa, P. V. Filevychb

a Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS Ukraine, L'viv, Ukraine
b L'viv National University of Veterinary Medicine and Biotechnology, L'viv, Ukraine

Abstract: It is proved, that for the majority (in the sense of probability measure) of functions $f$, analytic in the unit disk with unbounded Nevanlinna characteristic $T_f(r)$, and for all $\alpha<\beta\le\alpha+2\pi$ the relation
$$N_f(r,\alpha,\beta,0)\sim\frac{\beta-\alpha}{2\pi}T_f(r),\quad r\to1,$$
holds, where $N_f(r,\alpha,\beta,0)$ is the integrated counting functions of zeros of $f$ in the sector $ż\in\mathbb C\colon 0<|z|\le r, \alpha\le\arg_\alpha z<\beta\}$. The analogous proposition is obtained for entire functions under some conditions on their growth.

Keywords: analytic function, random analytic function, distribution of zeros, counting function, integrated counting function, Nevanlinna characteristic.

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UDC: 517.53

Citation: M. P. Mahola, P. V. Filevych, “The angular distribution of zeros of random analytic functions”, Ufimsk. Mat. Zh., 4:1 (2012), 122–135

Citation in format AMSBIB
\Bibitem{MahFil12} \by M.~P.~Mahola, P.~V.~Filevych \paper The angular distribution of zeros of random analytic functions \jour Ufimsk. Mat. Zh. \yr 2012 \vol 4 \issue 1 \pages 122--135 \mathnet{http://mi.mathnet.ru/ufa138} 

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