Zero separation results for solutions of second order linear differential equations

Ann. Acad. Sci. Fenn. Math.

Peláez

Rättyä

2014

Self Cite

Abstract. It is shown that for any positive, non-decreasing, continuous and unbounded doubling function ω on [0, 1), there exist two analytic infinite products f 0 and f 1 such that the asymptotic relation |f 0 (z)| + |f 1 (z)| ≍ ω(|z|) is satisfied for all z in the unit disc. It is also shown that both functions f j for j = 0, 1 satisfy T (r, f j ) ≍ log ω(r), as r → 1 − , and hence give examples of analytic functions for which the Nevanlinna characteristic admits the regular slow growth induced by ω.

Section: Introduction and Resultsmentioning

confidence: 99%

Section: Introduction and Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Jointly maximal products in weighted growth spaces

Ann. Acad. Sci. Fenn. Math.

Peláez

Rättyä

2014

Self Cite

“…By [28,Lemma 5.3] 5) where S a = {re iθ : |a| < r < 1, |θ − arg(a)| ≤ (1 − |a|)/2} denotes the Carleson square with respect to a ∈ D \ {0} and S 0 = D. See also [35,Lemma 3.4]. Solutions in VMOA, the closure of polynomials in BMOA, are discussed in Section 6 in which Theorem 3 is proved.…”

Section: Resultsmentioning

confidence: 99%

“…The necessary condition, corresponding to Nehari's theorem, was given by Kraus [22]. For more recent developments based on localization of the classical results, see [5]. In the case of higher order linear differential equations…”

Section: Introductionmentioning

confidence: 99%

Linear differential equations with slowly growing solutions

Trans. Amer. Math. Soc.

Huusko

Rättyä

2018

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Abstract. This research concerns coefficient conditions for linear differential equations in the unit disc of the complex plane. In the higher order case the separation of zeros (of maximal multiplicity) of solutions is considered, while in the second order case slowly growing solutions in H ∞ , BMOA and the Bloch space are discussed. A counterpart of the Hardy-Stein-Spencer formula for higher derivatives is proved, and then applied to study solutions in the Hardy spaces.

Converse growth estimates for ODEs with slowly growing solutions

2020

Math. Z.

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Let $$f_1,f_2$$ f 1 , f 2 be linearly independent solutions of $$f''+Af=0$$ f ′ ′ + A f = 0 , where the coefficient A is an analytic function in the open unit disc $${\mathbb {D}}$$ D of the complex plane $${\mathbb {C}}$$ C . It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function $$u=-\log \, (f_1/f_2)^{\#}$$ u = - log ( f 1 / f 2 ) # . For example, the case when $$\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 < \infty $$ sup z ∈ D | A ( z ) | ( 1 - | z | 2 ) 2 < ∞ and $$f_1/f_2$$ f 1 / f 2 is normal, is characterized by the condition $$\sup _{z\in {\mathbb {D}}} |\nabla u(z)|(1-|z|^2) < \infty $$ sup z ∈ D | ∇ u ( z ) | ( 1 - | z | 2 ) < ∞ . Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of u. Even if $$f_1,f_2$$ f 1 , f 2 are bounded linearly independent solutions of $$f''+Af=0$$ f ′ ′ + A f = 0 , it is possible that $$\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 = \infty $$ sup z ∈ D | A ( z ) | ( 1 - | z | 2 ) 2 = ∞ or $$f_1/f_2$$ f 1 / f 2 is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of $$f''+Af=0$$ f ′ ′ + A f = 0 in such a way that it solves an interpolation problem natural to bounded analytic functions, while $$|A(z)|^2(1-|z|^2)^3\, dm(z)$$ | A ( z ) | 2 ( 1 - | z | 2 ) 3 d m ( z ) remains to be a Carleson measure.