On non-normal solutions of linear differential equations

Gröhn, Janne

doi:10.1090/proc/13292

Cited by 7 publications

(11 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following result is an analogue of [10,Theorem 5], and is related to the classical 0, 1interpolation result due to Carleson [2,Theorem 2]. The Nevanlinna counterpart of Carleson's result is presented in Sect.…”

Section: Nevanlinna Interpolating Sequencesmentioning

confidence: 87%

“…It remains to show that all solutions of (1) belong to H ∞ p . On one hand, it is clear that f 1 ∈ H ∞ p by (10). On the other hand, (9) is a solution of (1) which is linearly independent to f 1 .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…2.5. Note that the coefficient condition A ∈ H ∞ 2 allows non-normal solutions by [10,Theorem 3] and [11,Theorem 1]; and even the normality of all solutions is not sufficient for A ∈ H ∞ 2 by Theorem 1(i) above. If f 1 , f 2 ∈ H ∞ are linearly independent solutions of (1) for A ∈ Hol(D), then A ∈ H ∞ 3 by a result of Steinmetz [44, p. 130].…”

Section: Bounded Solutionsmentioning

confidence: 99%

See 2 more Smart Citations

Converse growth estimates for ODEs with slowly growing solutions

Gröhn

2020

Math. Z.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Nevanlinna Interpolating Sequencesmentioning

confidence: 87%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Bounded Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

Converse growth estimates for ODEs with slowly growing solutions

Gröhn

2020

Math. Z.

Self Cite

View full text Add to dashboard Cite

show abstract

“…At least, each primitive g of an univalent function satisfies g ∈ N . Recently, similar normality considerations which have connections to differential equations, were done in [9]. If f ∈ U A loc and there exists 0 < δ < 1 such that f is univalent in each pseudohyperbolic disc ∆(a, δ) = {z ∈ D : |ϕ a (z)| < δ}, for a ∈ D, then f is called uniformly locally univalent.…”

Section: Distortion Theoremsmentioning

confidence: 99%

“…In this paper, we consider the growth condition 9) where 0 < C < ∞ is an absolute constant, for f ∈ U A loc . When (1.9) holds, we detect that f is univalent in horodiscs D(ae iθ , 1 − a), e iθ ∈ ∂D, for some a = a(C) ∈ [0, 1).…”

Section: Introductionmentioning

confidence: 99%

On Becker's univalence criterion

Huusko

Vesikko

2018

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Abstract. We study locally univalent functions f analytic in the unit disc D of the complex plane such that |f (z)/f (z)| (1 − |z| 2 ) ≤ 1 + C(1 − |z|) holds for all z ∈ D, for some 0 < C < ∞. If C ≤ 1, then f is univalent by Becker's univalence criterion. We discover that for 1 < C < ∞ the function f remains to be univalent in certain horodiscs. Sufficient conditions which imply that f is bounded, belongs to the Bloch space or belongs to the class of normal functions, are discussed. Moreover, we consider generalizations for locally univalent harmonic functions.

show abstract

Solutions of Complex Differential Equation Having Pre-given Zeros in the Unit Disc

Gröhn

2017

Constr Approx

Self Cite

View full text Add to dashboard Cite

Behavior of solutions of f ′′ + Af = 0 is discussed under the assumption that A is analytic in D and sup z∈D (1 − |z| 2 ) 2 |A(z)| < ∞, where D is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature.It is also proved that Λ ⊂ D is the zero-sequence of a non-trivial solution of f ′′ + Af = 0 where |A(z)| 2 (1 − |z| 2 ) 3 dm(z) is a Carleson measure if and only if Λ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.2 > 1 then non-trivial solutions may have infinitely many zeros [13]. The condition A ∈ H ∞ 2 is equivalent to the fact that zero-sequences of non-trivial solutions of (1) are separated with respect

show abstract

On non-normal solutions of linear differential equations

Cited by 7 publications

References 23 publications

Converse growth estimates for ODEs with slowly growing solutions

Converse growth estimates for ODEs with slowly growing solutions

On Becker's univalence criterion

Solutions of Complex Differential Equation Having Pre-given Zeros in the Unit Disc

Contact Info

Product

Resources

About