The angular value distribution of random analytic functions

Author M. P. Mahola, P. V. Filevych
marichka_stanko@ukr.net, filevych@mail.ru
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NASU, S. Z. Gzhytskyy Lviv National University of Veterinary Medicine and Biotechnologies

Abstract Let $\mathcal{R}\in(0,+\infty]$, $f(z)=\sum c_nz^n$ be an analytic function in the disk $\{z\colon |z|<\mathcal{R}\}$, $T_f(r)$ be the Nevanlinna characteristic, $N_f(r,\alpha,\beta,a)$ be the integrated counting function of $a$-points of $f$ in the sector $0<|z|\le r$, $\alpha\le\arg_{\alpha} z<\beta$, and $(\omega_n(\omega))$ be a~sequence of independent equidistributed on $[0,1]$ random variables. Under some conditions on the growth of $f$ it is proved that for random analytic function $f_\omega(z)=\sum e^{2\pi i\omega_n(\omega)}a_nz^n$ almost surely for every $a\in\mathbb{C}$ and all $\alpha<\beta\le\alpha+2\pi$ the relation $N_{f_\omega}(r,\alpha,\beta,a)\sim\frac{\beta-\alpha}{2\pi}T_{f_\omega}(r)$, $r\to\mathcal{R}$, holds outside some exceptional set $E\subset(0,\mathcal{R})$.
Keywords meromorphic function, value distribution; random analytic function; Nevanlinna characteristic; counting function; integrated counting function
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Pages 34-51
Volume 37
Issue 1
Year 2012
Journal Matematychni Studii
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