Some Estimates for the Solutions of the First Order Non-Algebraic Classes of Equations
- PDF / 495,207 Bytes
- 9 Pages / 612 x 792 pts (letter) Page_size
- 29 Downloads / 160 Views
and Complex Analysis
Some Estimates for the Solutions of the First Order Non-Algebraic Classes of Equations Barsegian Grigor1, 2* , Fanning Meng3, 1** , and Wenjun Yuan1*** 1 2
Guangzhou University, Guangzhou, China
Institute of Mathematics, National Academy of Science of Armenia, Yerevan, Armenia 3
School of Mathematics and Information Science, Guangzhou, China Received July 13, 2019; revised July 13, 2019; accepted February 6, 2020
Abstract—For some large classes of differential equations of the first order we give bounds for Ahlfors-Shimizu characteristics of meromorphic solutions in the complex plane of these equations. The considered equations largely generalize algebraic ones for which the obtained results imply the known Goldberg theorem. Characteristics of meromorphic solutions in a given domain weren’t studied at al. We consider solutions in a given domain of some (large) equations and give bounds for Ahlfors–Shimizu characteristic for these solutions. MSC2010 numbers : 34M05, 34M10, 34M99 DOI: 10.3103/S1068362320020053 Keywords: non-algebraic complex equations, solutions in a given domain, Ahlfors’ islands, Ahlfors characteristics.
In this paper, we consider complex differential equations of the first order in two cases: for meromorphic solutions in the complex plane and in a given domain. For algebraic equations, there is the classical Goldberg theorem related to meromorphic solutions in the complex plane. We study much larger equations; respectively our result implies as a particular case the mentioned Goldberg theorem. Characteristics of meromorphic solutions in a given domain weren’t studied at al. Recently G. Barsegian started similar studies; these studies were presented during his lectures in Guangzhou university in 2017. His aprroaches based on some new results related to arbitrary meromorphic functions in a given domain, see [4]. In this paper, we consider solutions in a given domain of some (large) equations and give bounds for Ahlfors–Shimizu characteristic for the solutions. 1. MEROMORPHIC SOLUTIONS IN THE COMPLEX PLANE We consider the following equation m m−1 + · · · + φm (z, w) = 0, (∗) φ0 (z, w) w + φ1 (z, w) w n(i) where φi (z, w) := μ(i)=1 ηi,μ(i) (z)χi,μ(i) (w) for i = 0, 1, 2, . . . , m and μ(i) = 1, 2, . . . , n(i). Obviously we should exclude the case φ0 (z, w) ≡ 0, since then the degree m reduces. We put the following restrictions: all coefficient χi,μ(i) (w) are meromorphic in C, all coefficients ηi,μ(i) (z) with i = 0 are entire functions and all coefficients η0,μ(0) (z) are polynomials. The equations (∗) with similar restrictions we will refer as (Fpe,m (C)). *
E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **
96
SOME ESTIMATES FOR THE SOLUTIONS
97
Notice that algebraic differential equations of the first order (see related studies in [8]) are particular cases of equations (Fpe,m (C)) when all mentioned above entire and meromorphic functions are polynomials. For meromorphic function w in C we make use of classical Ahlfors–Shimizu
Data Loading...
