A Survey on Blaschke-Oscillatory Differential Equations, with Updates

Heittokangas, Janne

doi:10.1007/978-1-4614-5341-3_3

Cited by 15 publications

(23 citation statements)

References 52 publications

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“…In this case it is clear that [17]. Second, by a more careful analysis on an example, which can be found in [10,13], we will prove that the solutions of (2) can have infinite uniformly separated and sparse zero sequences even if A ∈ B 1 2 . This result also shows that the requirement α ∈ (0, 1/2] in Theorem 4 is essential.…”

Section: Theorem 3 Let H(t) Be a Function As Above And Let {Z N } ∞mentioning

confidence: 90%

“…Let B be an interpolating Blaschke product having zeros at the points z n satisfying (11). As in the proof of Theorem 2, we can find a function g ∈ H(‫)ބ‬ such that f = Be g solves (2), where A(z) is given by (10) r) . …”

Section: Lemma 3 Let B and H Be As In Lemma 2 And In Addition Supmentioning

confidence: 99%

“…Next we will widen our point of view to concern all solutions. First, an example of a nonoscillatory differential equation of the form (2) is constructed in [10,Section 4.3] such that all nontrivial solutions of (2) are of unbounded characteristic. In this case it is clear that [17].…”

Section: Theorem 3 Let H(t) Be a Function As Above And Let {Z N } ∞mentioning

confidence: 99%

“…We have the following partial result. THEOREM 1 ( [10], Theorem 19). Let {z n } ∞ n=1 be a uniformly separated sequence of nonzero points in ‫ބ‬ satisfying the condition (8) for some α ∈ (0, 1).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Complexity of Finding a Necessary and Sufficient Condition for Blaschke-Oscillatory Equations

Heittokangas

Reijonen

2014

Glasgow Math. J.

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If A(z) belongs to the Bergman space B 1 2 , then the differential equation f + A(z)f = 0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to B 1 2 . It is demonstrated that certain "nice" Blaschke sequences can be zero sequences of solutions in both cases when A ∈ B 1 2 or A ∈ B 1 2 . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ B 1 2 .

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Section: Theorem 3 Let H(t) Be a Function As Above And Let {Z N } ∞mentioning

confidence: 90%

Section: Lemma 3 Let B and H Be As In Lemma 2 And In Addition Supmentioning

confidence: 99%

Section: Theorem 3 Let H(t) Be a Function As Above And Let {Z N } ∞mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Complexity of Finding a Necessary and Sufficient Condition for Blaschke-Oscillatory Equations

Heittokangas

Reijonen

2014

Glasgow Math. J.

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“…as the spherical length of w(γ(z 1 , z 2 )) is uniformly bounded from below. Therefore, (11) implies that |z 1 − z 2 | is uniformly bounded away from zero.…”

Section: Proofs Of Theorems 6 Andmentioning

confidence: 98%

Slowly growing solutions of ODEs revisited

Gröhn¹

2018

Ann. Acad. Sci. Fenn. Math.

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Solutions of the differential equation f ′′ + Af = 0 are considered assuming that A is analytic in the unit disc D and satisfiesBy recent results in the literature, such restriction has been associated to coefficient conditions which place all solutions in the Bloch space B.In this paper it is shown that any coefficient condition implying (⋆) fails to detect certain cases when Bloch solutions do appear. The converse problem is also addressed: What can be said about the growth of the coefficient A if all solutions of f ′′ + Af = 0 belong to B? An overall revised look into slowly growing solutions is presented, emphasizing function spaces B, BMOA and VMOA.

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

Gröhn

2017

Constr Approx

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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

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A Survey on Blaschke-Oscillatory Differential Equations, with Updates

Cited by 15 publications

References 52 publications

On the Complexity of Finding a Necessary and Sufficient Condition for Blaschke-Oscillatory Equations

On the Complexity of Finding a Necessary and Sufficient Condition for Blaschke-Oscillatory Equations

Slowly growing solutions of ODEs revisited

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

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