The Carleson measure and meromorphic functions of uniformly bounded characteristic

Pavićević, Žarko

doi:10.5186/aasfm.1991.1619

Cited by 5 publications

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“…This completes the proof of (ii), as f 1 / f 2 is a normal function in the Nevanlinna class if and only if the right-hand side of (27) is finite [38,Theorem 1].…”

Section: Proofs Of Theorem 6 and Propositionmentioning

confidence: 65%

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Converse growth estimates for ODEs with slowly growing solutions

Gröhn

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.

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“…This completes the proof of (ii), as f 1 / f 2 is a normal function in the Nevanlinna class if and only if the right-hand side of (27) is finite [38,Theorem 1].…”

Section: Proofs Of Theorem 6 and Propositionmentioning

confidence: 65%

“…A meromorphic function w is said to be of uniformly bounded characteristic, that is w ∈ UBC, if w # (z) 2 (1 − |z| 2 ) dm(z) is a Carleson measure. We refer to [38,Theorem 3] for more details.…”

Section: Blaschke-oscillatory Equationsmentioning

confidence: 99%

Converse growth estimates for ODEs with slowly growing solutions

Gröhn

2020

Math. Z.

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“…This completes the proof of (ii), as f 1 /f 2 is a normal function in the Nevanlinna class if and only if the right-hand side of ( 27) is finite [36,Theorem 1].…”

Section: Proofs Of Theorem 6 and Propositionmentioning

confidence: 65%

“…then e −Λ̺ω(z From this estimate we deduce (36). Assume that (36) holds for some constant 0 < Λ < ∞. Fix z 2 ∈ D. Since lim…”

Section: Proofs Of Theorem 13 and Corollary 15mentioning

confidence: 87%

Converse growth estimates for ODEs with slowly growing solutions

Gröhn¹

2018

Preprint

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Let f1, f2 be linearly independent solutions of f ′′ + Af = 0, where the coefficient A is an analytic function in the open unit disc D of C. It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function u = − log (f1/f2) # . For example, the case whenDifferent types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of u.Even if f1, f2 are bounded linearly independent solutions of f ′′ + Af = 0, it is possible that sup z∈D |A(z)|(1 − |z| 2 ) 2 = ∞ or f1/f2 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 results 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 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) remains to be a Carleson measure.

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“…This corrects Proposition 2 (i) in [4] where "equality" between Q # p and M # p was "proved". The proof of Proposition 2 (i) was based on [9,Corollary]. The following theorem shows very strongly that some results for the spaces of analytic functions do not remain true for the corresponding classes of meromorphic functions.…”

Section: −āZmentioning

confidence: 99%