Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(2): 419-431

Online first article September 7, 2020      Printed March 31, 2021

https://doi.org/10.4134/BKMS.b200321

Copyright © The Korean Mathematical Society.

The infinite growth of solutions of second order linear complex differential equations with completely regular growth coefficient

Guowei Zhang

Anyang Normal University

Abstract

In this paper we discuss the classical problem of finding conditions on the entire coefficients $A(z)$ and $B(z)$ guaranteeing that all nontrivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order. We assume $A(z)$ is an entire function of completely regular growth and $B(z)$ satisfies three different conditions, then we obtain three results respectively. The three conditions are (1) $B(z)$ has a dynamical property with a multiply connected Fatou component, (2) $B(z)$ satisfies $T(r,B)\sim \log M(r,B)$ outside a set of finite logarithmic measure, (3) $B(z)$ is extremal for Denjoy's conjecture.

Keywords: Entire function, infinite order, complex differential equation

MSC numbers: 30D35, 34M10, 37F10

Supported by: This work was supported by NSFC(no.11426035), the key scientific research project for higher education institutions of Henan Province, China (no. 18A110002) and training program for young backbone teachers of colleges and universities in Henan Province, China (no. 2017GGJS126)