Zero distribution of solutions of complex linear differential equations determines growth of coefficients

Heittokangas, Janne; Rättyä, Jouni

doi:10.1002/mana.201010038

Cited by 10 publications

(23 citation statements)

References 13 publications

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“…has only one dominant term (highest-degree term), then (1.1) has no admissible transcendental meromorphic solutions with a few poles. There are also many other papers concerning the structure of solutions to various differential equations (for example [1,4,[10][11][12][13][14]). In this paper, we consider the algebraic differential equation…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Solutions to Nonhomogeneous Algebraic Differential Equations and Their Application

Liao¹,

2014

J. Aust. Math. Soc.

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We consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then $$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$ With this in hand, we also prove that if $f$ is a transcendental entire function, then $f'p_k(f)+q_m(f)$ assumes every complex number $\alpha $, with one possible exception, infinitely many times, where $p_k(f), q_m(f)$ are polynomials in $f$ with degrees $k$ and $m$ with $k\ge m+1$. This result generalizes a theorem originating from Hayman [‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2)70(2) (1959), 9–42].

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On Solutions to Nonhomogeneous Algebraic Differential Equations and Their Application

Liao¹,

2014

J. Aust. Math. Soc.

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“…For example, the choice E(z) = , j = 1, 2, then reduce to constant multiples of the functions f 1 , f 2 in (2.9), respectively. We have seen that it is possible to construct zero-free solutions bases {f 1 , f 2 } for (1.1) in the cases of C and D. It has recently been proved [14] in both cases that arbitrary linear combinations f = C 1 f 1 + C 2 f 2 typically have the maximal quantity of zeros when compared to the growth of f 1 , f 2 .…”

Section: Concluding Remarks On Zero-free Solution Basesmentioning

confidence: 99%

A unit disc analogue of the Bank-Laine conjecture does not hold

Heittokangas

Tohge

2011

Ann. Acad. Sci. Fenn. Math.

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where f 1 , f 2 are linearly independent solutions of f + A(z)f = 0 and λ(g) stands for the exponent of convergence of the zeros of g. This conjecture has been verified in the case ρ(A) ≤ 1/2, while counterexamples have been found in the cases ρ(A) ∈ N∪{∞}. The aim of this paper is to illustrate that no growth condition on A(z) alone yields a unit disc analogue of the Bank-Laine conjecture. The main discussion yields solutions to two open problems recently stated by Cao and Yi.

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“…. Therefore, to avoid unnecessary repetition, we merely sketch a proof of (a), and refer to [10] for a further discussion on the topic. This is not a surprise, because the growth of g is determined via g = h −k by a solution h of (1·3) with B entire.…”

Section: Functions Of Maximal Growth In αmentioning

confidence: 99%

“…, k −1, see [10] for details. Therefore one implication in Theorem B(a) follows by the growth estimates for the solutions of (1·8), see Lemma D(a) below.…”

Section: It Is Well Known That λ(H) σ (H) For All H ∈ H(d)mentioning

confidence: 99%

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Generalized Schwarzian derivatives and higher order differential equations

Chuaqui

Gröhn

Rättyä

2011

Math. Proc. Camb. Phil. Soc.

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It is shown that the well-known connection between the second order linear differential equation h + B(z) h = 0, with a solution base {h 1 , h 2 }, and the Schwarzian derivativeThis generalization depends upon an appropriate definition of the generalized Schwarzian derivative S k ( f ) of a function f which is induced by k −1 ratios of linearly independent solutions of h (k) + B(z) h = 0. The class R k ( ) of meromorphic functions f such that S k ( f ) is analytic in a given domain is also completely described. It is shown that if is the unit disc D or the complex plane C, then the order of growth of f ∈ R k ( ) is precisely determined by the growth of S k ( f ), and vice versa. Also the oscillation of solutions of h (k) + B(z) h = 0, with the analytic coefficient B in D or C, in terms of the exponent of convergence of solutions is briefly discussed.

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Zero distribution of solutions of complex linear differential equations determines growth of coefficients

Cited by 10 publications

References 13 publications

On Solutions to Nonhomogeneous Algebraic Differential Equations and Their Application

On Solutions to Nonhomogeneous Algebraic Differential Equations and Their Application

A unit disc analogue of the Bank-Laine conjecture does not hold

Generalized Schwarzian derivatives and higher order differential equations

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