The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation

Guowei Zhang; Guowei Zhang

doi:10.3934/math.20231028

AIMS Mathematics

2023, Volume 8, Issue 9: 20169-20186. doi: 10.3934/math.20231028

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Research article

The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation

Guowei Zhang ^,

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China

Received: 03 May 2023 Revised: 24 May 2023 Accepted: 29 May 2023 Published: 19 June 2023
MSC : 30D05, 37F10, 37F50

In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between $ T (r, f) $ and $ \log M (r, f) $ when the modulus of variable $ |z| = r $ is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results.
- Petrenko's deviation,
- Julia set,
- entire function,
- complex equation
Citation: Guowei Zhang. The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation[J]. AIMS Mathematics, 2023, 8(9): 20169-20186. doi: 10.3934/math.20231028

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Abstract

In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between $ T (r, f) $ and $ \log M (r, f) $ when the modulus of variable $ |z| = r $ is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results.

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