On growth order of solutions of differential equations in a neighborhood of a branch point

Author A. Z. Mokhonko, A. A. Mokhonko
mohon@yandex.ru
National University ``Lvivska Politechnika''

Abstract Let $M_k$ be {the} set of $k$-valued meromorphic in $G=\{z\colon r_0\leqslant |z|\}$ functions with {a}~branch point of order $k-1$ {at} $\infty$; let $E_\ast$ be a set of circles {with finite} sum of radii. Denote $ M_\ast(r,f)=\max|f(z)|,\ z\in\{te^{i\theta}\colon 0\leqslant\theta\leqslant2k\pi,\ r_0\leqslant t\leqslant r\}\setminus E_\ast,\ f\!\in\! M_k; $ $m(r,f)=\frac{1}{2\pi k}\int_0^{2\pi k}\!\ln^+\!|f(re^{i\theta})|d\theta$. If $f\in M_k$ is a solution of the equation $P(z,f,f')=0$ and $P$ is a polynomial in all variables then either $|f(re^{i\theta})|\le r^\nu,$ $re^{i\theta}\in G\setminus E_\ast,\ \nu>0$ or $m(r,f)$ has growth order $\rho\geqslant\frac{1}{2k}$, and the following equality holds $\ln M_\ast(r,f)=(c+o(1))r^\rho,$ $c\neq0,$ $r\to+\infty.$
Keywords algebraic differential equation; branch point of analytic function; meromorphic solution; order of growth
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Pages 53-65
Volume 40
Issue 1
Year 2013
Journal Matematychni Studii
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