Solutions of \(f''+A (z) f=0\) in the unit disc having Blaschke sequences as the zeros
type: Item
instance of: scholarly article
title: Solutions of \(f''+A (z) f=0\) in the unit disc having Blaschke sequences as the zeros
zbMATH Open document ID: 1099.34076
DOI: 10.1007/BF03321085
author: Janne Heittokangas
published in: Computational Methods and Function Theory
publication date: 2006-03-10T00:00:00Z
review text: The zero sequences of the nontrivial solutions of \(f''+ A(z)f= 0\) are studied, where \(A(z)\) is analytic in \(D= \{z:|z|< 1\}\). The author shows that if \(A(z)\) is analytic in \(D\) satisfying at least one of the conditions \[ \iint_D|A(z)|^{1/2} d\sigma<\infty\quad\text{and}\quad \iint_D |A(z)|(1-|z|)\,d\sigma< \infty, \] then the equation is Blaschke-oscillatory. On the other hand, he obtains the following result. Let \(\{z_n\}\) be an infinite interpolating Blaschke sequence of nonzero points in \(D\). Then, the following assertions hold. (1) For each function \(F\), analytic in \(D\), there exists a function \(A_F(z)\), analytic in \(D\) and depending on \(F\), such that \(f''+ A_F(z)f= 0\) possesses a solution \(f_F\) having zeros precisely at the points \(z_n\). (2) There are uncountably many functions \(F\), analytic in \(D\), each resulting in a different function \(A_F(z)\) satisfying the properties stated in (1).
reviewed by: Zhongqiu Ye
Mathematics Subject Classification ID: 34M10
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 5011335
zbMATH Keywords: Blaschke sequences
zbMATH Keywords: zeros of solutions
zbMATH Keywords: zero distribution
zbMATH Keywords: zero sequences
MaRDI profile type: Publication
cites work: A note on the zero-sequences of solutions of linear differential equations
cites work: On the Oscillation Theory of f � � + Af = 0 where A is Entire
cites work: Q5618011
cites work: Q4490703
cites work: GENERALIZED LOGARITHMIC DERIVATIVE ESTIMATES OF GOL'DBERG–GRINSHTEIN TYPE
cites work: Q3370109
cites work: Remarks on a paper by Zeev Nehari
cites work: Integral means for the \(n\)'th derivative of Blaschke products.
cites work: Nevanlinna theory and complex differential equations
cites work: \(H^p\)-derivatives of Blaschke products
cites work: On the zeros of the solutions of \(w''(z)+p(z)w(z) = 0\)
cites work: The Schwarzian derivative and schlicht functions
cites work: On the Zeros of Solutions of Second-Order Linear Differential Equations
cites work: Interpolation in H p -Spaces
cites work: On the mean growth of the solutions of complex linear differential equations in the disk
cites work: A note on the zero-sequences of solutions of 𝑓”+𝐴𝑓=0
cites work: Complex Nonoscillation Theorems and Criteria of Univalence
cites work: Construction of a Differential Equation y � � + Ay = 0 with Solutions having the Prescribed Zeros
cites work: Nontangential Interpolating Sequences and Interpolation by Normal Functions
description: scientific article; zbMATH DE number 5011335