Asymptotic behaviour of pth means of analytic and subharmonic functions in the unit disc and angular distribution of zeros

Chyzhykov, Igor

doi:10.1007/s11856-020-1996-x

Cited by 8 publications

(4 citation statements)

References 21 publications

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“…In general, σ M (f ) ≤ ρ ∞ (f ) for all analytic functions in D. Observe that the left-hand inequality in (1.2) is no longer true if σ M (f ) < 1. A Blaschke product B such that ρ p (B) = 1 − 1 p while σ M (B) = 0 gives a counter example [6]. Further properties of the order ρ ∞ (f ) can be found in [5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Generalization of proximate order and applications

Chyzhykov¹,

Filevych²,

Rättyä³

2020

Preprint

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We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem of a quasi proximate order, i.e. a counterpart of Valiron's theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright and C. N. Linden on asymptotic behavior of analytic functions in the unit disc.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Generalization of proximate order and applications

Chyzhykov¹,

Filevych²,

Rättyä³

2020

Preprint

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“…I. Lyubarskii studied the decomposition of functions with a triangle indicator diagram ( [10]). I. E. Chyzhykov researched similar issues in terms of subharmonic functions ( [11]). This problem arises as a result of studies of the completeness function ( [12]) and is interesting in the theory of integral operators, shift operator ( [13]) and convolution type equations.…”

Section: Definitionmentioning

confidence: 99%

Hilbert transform on $W_{\sigma}^1$

Dilnyi

Voitovych

2019

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“…Asymptotic behavior of M p (r, u) where 1 < p < ∞ is described in [5], see also [8]. The disadvantage of the mentioned results is that they do not describe local growth of a harmonic function.…”

Section: An Applicationmentioning

confidence: 99%

Estimates of conjugate harmonic functions with given set of singularities with application

Chyzhykov,

Kosanyak

2019

Preprint

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Let E be an arbitrary closed set on the unit circle ∂D, u be a harmonic function on the unit disk D satisfying |u(z)| (1 − |z|) γ ρ −q (z) where ρ(z) = dist(z, E), γ, q are some real constants, γ ≤ q. We establish an estimate of the conjugate ũ of the same type which is sharp in some sense and in the case E = ∂D coincides with known estimates. As an application we describe growth classes defined by the non-radial condition |u(z)| ρ −q (z) in terms of smoothness of the Stieltjes measure associated to the harmonic function u.

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Asymptotic behaviour of pth means of analytic and subharmonic functions in the unit disc and angular distribution of zeros

Cited by 8 publications

References 21 publications

Generalization of proximate order and applications

Generalization of proximate order and applications

Hilbert transform on $W_{\sigma}^1$

Estimates of conjugate harmonic functions with given set of singularities with application

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