On interpolating Blaschke products and Blaschke-oscillatory equations
type: Item
instance of: scholarly article
title: On interpolating Blaschke products and Blaschke-oscillatory equations
zbMATH Open document ID: 1227.30041
author: Janne Heittokangas
published in: Constructive Approximation
publication date: 2011-08-16T00:00:00Z
review text: This research is partially a continuation of the author's paper [``Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class'', Kodai Math. J. 30, No. 2, 263--279 (2007; Zbl 1134.30025)]. If \(\{z_n\}\) is an infinite sequence of nonzero points in the unit disc \(D = \{z: |z| <1\}\) satisfying \(\sum_{n=1}^\infty(1-|z_n|)<\infty\), then the Blaschke product \(B(z)=\prod_{ n=1}\frac{|z_n|}{z_n}\frac{z_n-z}{1-\overline z_nz}\) represents an analytic function in \(D\) and has zeros precisely at the points \(z_n\). The next result relates the strengthened Blaschke condition given by \[ \sum_{n=1}(1-|z_n|)^\alpha<\infty,\quad \alpha\in(0,1],\tag{1} \] to logarithmic derivatives of Blaschke products. Let \(B\) be a Blaschke product with zeros \(z_n\neq0\) such that \text{(1)} holds for some \(\alpha\in (0,1]\), and let \(k\in\mathbb N\). Suppose that \(\omega : [1,\infty)\to [1,\infty)\) is a continuous and increasing function such that \[ \int_1^\infty\frac{x}{\omega(x)}\,dx<\infty. \] Then there exists a set \(E_1\subset [0,1)\) which satisfies \[ \int_{E_1}\frac{dr}{1-r}<\infty, \] such that for all \(z\in D\) satisfying \(|z|\notin E_1\), we have \[ \left|\frac{B^{(k)}(z)}{B(z)}\right|=O\left(\left(\frac{1}{1-|z|}\right)^{(1+\alpha)k}\omega\left(\log\frac{e}{1-|z|}\right)^k\right). \] A sequence \(\{z_n\}\) of points in \(D\) is called uniformly separated provided that \[ \delta:=\inf_{k}\prod_{ n\neq k}\left|\frac{z_n-z_k}{1-\overline z_nz_k}\right|>0, \] and is called exponential if there is a constant \(q\in (0,1)\) such that \[ 1-|z_{n+1}|\leq q(1-|z_n|). \] Such Blaschke products are known as interpolating Blaschke products which appear naturally in studying the frequency of zeros of solutions of \(f'' + A(z)f = 0\), where \(A(z)\) is analytic in \(D\). In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a …
Mathematics Subject Classification ID: 30J10
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 5939413
zbMATH Keywords: Blaschke product
zbMATH Keywords: logarithmic derivative
zbMATH Keywords: interpolating sequence
zbMATH Keywords: uniformly separated sequence
zbMATH Keywords: exponential sequence
zbMATH Keywords: prescribed zero sequence
zbMATH Keywords: oscillation theory
zbMATH Keywords: Blaschke-oscillatory equation
zbMATH Keywords: frequency of zeros
MaRDI profile type: Publication
full work available at URL: s00365-010-9100-0
OpenAlex ID: W1988817811
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DOI: 10.1007/S00365-010-9100-0
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description: scientific article; zbMATH DE number 5939413