On the theorem of tumura-clunie

Reyzl, I.

doi:10.1080/17476939508814847

Cited by 4 publications

(3 citation statements)

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“…Since [4], many theorems of Tumura-Clunie type have appeared in the literature, see [14,16,23,24] and the references therein. Typical applications of these theorems are among differential polynomials and non-linear differential equations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Meromorphic solutions of some linear functional equations

et al. 2000

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Meromorphic solutions of non-linear differential equations of the form f 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 α1(z) + p 2 (z)e α2 (z) , where p 1 , p 2 are small functions of f and α 1 , α 2 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 ′′ + r 1 (z)h ′ + r 0 (z)h = r 2 (z) with rational coefficients r 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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Meromorphic solutions of some linear functional equations

et al. 2000

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“…Since [5], many theorems of Tumura–Clunie type have appeared in the literature, see [16, 18, 25, 26] and the references therein. Typical applications of these theorems are among differential polynomials and non‐linear differential equations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Heittokangas

Latreuch

Wang

et al. 2021

Mathematische Nachrichten

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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.

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“…Theorem 1.1 is an extension of Tumura-Clunie theory which is based on a theorem proposed by Tumura [15], but whose proof was completed by Clunie [4]. Since then, the non-linear differential equation (1.1) has been studied extensively over the years, see [13,14,19,20] and the references therein.…”

Section: Introductionmentioning

confidence: 99%