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On non-separated zero sequences of solutions of a linear differential equation

Part of: Qualitative theory Function theory on the disc Geometric function theory Spaces and algebras of analytic functions

Published online by Cambridge University Press:  30 April 2021

Igor Chyzhykov  and
Jianren Long
Igor Chyzhykov
Affiliation:
Faculty of Mechanics and Mathematics, Lviv Ivan Franko National University, Universytets'ka 1, 79000Lviv, Ukraine ([email protected])
Jianren Long
Affiliation:
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, Guizhou, China ([email protected])
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Abstract

Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb {D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb {D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ for any $p > 0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.

Keywords

interpolationunit discanalytic functionoscillation of solutiondifferential equationprescribed zeros

MSC classification

Primary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory
Secondary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) 30H99: None of the above, but in this section 30J99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 247 - 261
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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