Zur Wertverteilung von Exponentialpolynomen

Steinmetz, Norbert

doi:10.1007/bf01167971

Cited by 58 publications

(58 citation statements)

References 6 publications

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“…This is apparent by the proof of [18,Satz 4], and has also been stated in [19, p. 462]. Divide (3.8) by r ρ(A) and let r → ∞.…”

Section: By Continuity Again We Havementioning

confidence: 88%

“…As described in [18], g can be written in the normalized form (1 + deg(P j ))n l−j with polynomial coefficients [22]. Here n = max j {deg(Q j )}.…”

Section: Proofmentioning

confidence: 99%

“…A few things from [18] need to be reviewed. The convex hull co(W ) of a finite set W ⊂ C is the intersection of finitely many closed half-planes each containing W .…”

Section: Proofmentioning

confidence: 99%

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Completely regular growth solutions of second order complex linear differential equations

Heittokangas¹,

Laine²,

Tohge³

et al. 2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. If A(z) and B(z) are transcendental entire functions, then all solutions of the differential equation f ′′ + A(z)f ′ + B(z)f = 0 are entire and typically of infinite order. Simple examples show that finite order solutions are also possible. Assuming that A(z) and B(z) are of completely regular growth, Gol'dberg-Ostrovskii-Petrenko asked whether all solutions of finite order are of completely regular growth also. This problem remains unsolved, but several aspects of the problem are addressed. Exponential polynomials form an important subclass of functions of completely regular growth, and they are known to always satisfy certain linear differential equations. Hence cases where the coefficients and/or the solutions are exponential polynomials are also discussed.

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“…This is apparent by the proof of [18,Satz 4], and has also been stated in [19, p. 462]. Divide (3.8) by r ρ(A) and let r → ∞.…”

Section: By Continuity Again We Havementioning

confidence: 88%

“…As described in [18], g can be written in the normalized form (1 + deg(P j ))n l−j with polynomial coefficients [22]. Here n = max j {deg(Q j )}.…”

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

Completely regular growth solutions of second order complex linear differential equations

Heittokangas¹,

Laine²,

Tohge³

et al. 2015

Ann. Acad. Sci. Fenn. Math.

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“…The proof of (e), to be carried out in Sections 4-5, is more involved forming the core of this paper. The key idea is to use the value distribution results for exponential polynomials due to Steinmetz [4].…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, following the reasoning in[4], suppose that the polynomials Q j (z) in (1.6) are pairwise different and normalized by Q j (0) = 0. The representation (1.6) is then uniquely determined, and the functions P j (z)e Q j (z) are linearly independent.…”

mentioning

confidence: 99%

Exponential polynomials as solutions of certain nonlinear difference equations

Wen

Heittokangas

Lain

2012

Acta. Math. Sin.-English Ser.

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Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form f n + L(z, f ) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f (z) 2 + q(z)f (z + 1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f (z) n + q(z)e Q(z) f (z + c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.

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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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Zur Wertverteilung von Exponentialpolynomen

Cited by 58 publications

References 6 publications

Completely regular growth solutions of second order complex linear differential equations

Completely regular growth solutions of second order complex linear differential equations

Exponential polynomials as solutions of certain nonlinear difference equations

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

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