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

Abstract

Meromorphic solutions of non‐linear differential equations of the form fn+Pfalse(z,ffalse)=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 hfalse(zfalse)=p1false(zfalse)eα1false(zfalse)+p2false(zfalse)eα2false(zfalse), 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′′+r1false(zfalse)h′+r0false(zfalse)h=r2false(zfalse) 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.

Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

“…Next we discuss results on Riccati equations [5], Malmquist type equations [35], and Tumura-Clunie type equations [52,113,114]. The reason why we focus on these papers is that each of them contains a very nice reference list on the corresponding study and provides with many interesting examples in terms of exponential sums.…”

“…where n ≥ 2 is an integer, h is a meromorphic function in C, and P (z, f ) is a differential polynomial in f and its derivatives with small meromorphic coefficients (compared to f ). The name for these equations arises from a theorem of Tumura published in 1937 the proof of which was completed by Clunie in 1962 [52]. The special case of (4.27) of the form…”

“…The known solutions of the general equation (4.27) are typically exponential polynomials with rational coefficients, see [52] and the references therein. Apart from exponential polynomials, entire solutions of (4.27) of every half-integer order also exist [52]. In these examples, the function h is a solution of a second order linear differential equation with rational coefficients.…”

Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

“…where 𝑛 ⩾ 2 is an integer, ℎ is a meromorphic function in ℂ, and 𝑃(𝑧, 𝑓) is a differential polynomial in 𝑓 and its derivatives with small meromorphic coefficients (compared to 𝑓). The name for these equations arises from a theorem of Tumura published in 1937 the proof of which was completed by Clunie in 1962 [54]. The special case of (4.27) of the form…”

“…Next we discuss results on Riccati equations [5], Malmquist type equations [37], and Tumura–Clunie type equations [54, 115, 116]. The reason why we focus on these papers is that each of them contains a very nice reference list on the corresponding study and provides with many interesting examples in terms of exponential sums.…”