The angular value distribution of analytic and random analytic functions (in Ukrainian)

Author M. P. Mahola, P. V. Filevych
marichka_stanko@ukr.net, filevych@mail.ru
²íñòèòóò ïðèêëàäíèõ ïðîáëåì ìåõàí³êè ³ ìàòåìàòèêè ³ì. ß. Ñ. ϳäñòðèãà÷à ÍÀÍ Óêðà¿íè, Ïðèêàðïàòñüêèé íàö³îíàëüíèé óí³âåðñèòåò ³ì. Â. Ñòåôàíèêà

Abstract Let $a\in\mathbb{C}$, $f(z)=\sum c_nz^n$ be an analytic function in the disk $|z|<1$ such that $S_f(r){:=}\left(\sum |c_n|^2r^{2n}\right)^{\frac12}\to+\infty$, $r\to1$, $N_f(r,\alpha,\beta,a)$ be the integrated counting function of $a$-points of $f$ in the sector $0<|z|\le r$, $\alpha\le\arg_{\alpha} z<\beta$, $N_f(r,a)=N_f(r,0,2\pi,a)$, and $E\subset(0,1)$ be a set such that $\sup E=1$. It is proved that if $N_{f}(r,a)\sim\ln S_{f}(r)$, $E\ni r\to1$, then $N_{f}(r,\alpha,\beta,a)\sim\frac{\beta-\alpha}{2\pi}\ln S_{f}(r)$, $E\ni r\to1$, uniformly in $\alpha<\beta\le\alpha+2\pi$.
Keywords value distribution, analytic function; random analytic function; integrated counting function; angular integrated counting function
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Pages 147-153
Volume 38
Issue 2
Year 2012
Journal Matematychni Studii
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