On meromorphic solutions of non‐linear differential equations of Tumura–Clunie type

J. Heittokangas, Z. Latreuch, Jessie Wang and M. A. Zemirni

Mathematische Nachrichten, 2021, vol. 294, issue 4, 748-773

Abstract: Meromorphic solutions of non‐linear differential equations of the form fn+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)=p1(z)eα1(z)+p2(z)eα2(z), where p1,p2 are small functions of f and α1,α2 are 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 equation h′′+r1(z)h′+r0(z)h=r2(z) with rational coefficients r0,r1,r2, 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.

Date: 2021
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