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On meromorphic solutions of non‐linear differential equations of Tumura–Clunie type
Author(s) -
Heittokangas J.,
Latreuch Z.,
Wang J.,
Zemirni M. A.
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201900120
Subject(s) - meromorphic function , mathematics , order (exchange) , linear differential equation , rational function , differential equation , type (biology) , integer (computer science) , entire function , polynomial , function (biology) , mathematical analysis , pure mathematics , ecology , finance , evolutionary biology , computer science , economics , biology , programming language
Meromorphic solutions of non‐linear differential equations of the formf n + P ( z , f ) = h are investigated, where n ≥ 2 is an integer, h is a meromorphic function, and P ( z , f ) is differential polynomial in f and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when h has the particular form h ( z ) = p 1 ( z ) e α 1 ( z )+ p 2 ( z ) e α 2 ( z ), wherep 1 , p 2are small functions of f andα 1 , α 2are entire functions. In such a case the order of h is either a positive integer or equal to infinity. In this article it is assumed that h is a meromorphic solution of the linear differential equationh ′ ′ + r 1 ( z ) h ′ + r 0 ( z ) h = r 2 ( z )with rational coefficientsr 0 , r 1 , r 2 , and hence the order of h is a rational number. Recent results by Liao–Yang–Zhang (2013) and Liao (2015) follow as special cases of the main results.