zbMATH — the first resource for mathematics

Blaschke-oscillatory equations of the form \(f''+A(z)f=0\). (English) Zbl 1094.34060
Summary: We study the zero sequences of the nontrivial solutions of
\[ f''+A(z)f=0, \tag† \] where \(A(z)\) is analytic in the unit disc. We offer several aspects illustrating the fact that it is not so uncommon for these zero sequences to be Blaschke sequences. The typical results can be divided into two categories: (1) We search for conditions on \(A(z)\) under which the zero sequences of solutions of \((†)\) are Blaschke sequences. (2) For given Blaschke sequences satisfying certain conditions, we construct an analytic function \(A(z)\) (of minimal growth) such that these Blaschke sequences are the zero sequences of certain solutions of \((†)\).
This discussion is a continuation of the recent paper of the author [Comput. Methods Funct. Theory 5, 49–63 (2005; Zbl 1099.34076)].

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
Full Text: DOI
[1] Bank, S.; Laine, I., On the oscillation theory of \(f'' + A f = 0\), where A is entire, Trans. amer. math. soc., 273, 351-363, (1982) · Zbl 0505.34026
[2] Bank, S.; Laine, I., On the zeros of meromorphic solutions and second-order linear differential equations, Comment. math. helv., 58, 656-677, (1983) · Zbl 0532.34008
[3] Duren, P., Theory of \(H^p\) spaces, (1970), Academic Press New York · Zbl 0215.20203
[4] Linden, C.N., \(H^p\) derivatives of Blaschke products, Michigan math. J., 23, 43-51, (1976) · Zbl 0334.30017
[5] Heittokangas, J., Solutions of \(f'' + A(z) f = 0\) in the unit disc having Blaschke sequences as the zeros, Comput. methods funct. theory, 5, 49-63, (2005) · Zbl 1099.34076
[6] J. Heittokangas, Growth estimates for logarithmic derivatives of Blaschke products, preprint, 2005
[7] Heittokangas, J.; Korhonen, R.; Rättyä, J., Generalized logarithmic derivative estimates of gol’dberg – grinshtein type, Bull. London math. soc., 36, 105-114, (2004) · Zbl 1067.30060
[8] J. Heittokangas, R. Korhonen, J. Rättyä, Linear differential equations with coefficients in weighted Bergman and Hardy spaces, preprint, 2004
[9] J. Heittokangas, I. Laine, Solutions of \(f'' + A(z) f = 0\) with prescribed sequences of zeros, Acta Math. Univ. Comenian., in press · Zbl 1164.34562
[10] Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. amer. math. soc., 55, 545-551, (1949) · Zbl 0035.05104
[11] Pommerenke, Ch., On the Mean growth of the solutions of complex linear differential equations in the disk, Complex variables, 1, 23-38, (1982) · Zbl 0464.34010
[12] Protas, D., Blaschke products with derivative in \(H^p\) and \(B^p\), Michigan math. J., 20, 393-396, (1973) · Zbl 0258.30032
[13] Rudin, W., The radial variation of analytic functions, Duke math. J., 22, 235-242, (1955) · Zbl 0064.31105
[14] Shen, L.C., Construction of a differential equation \(y'' + A y = 0\) with solutions having the prescribed zeros, Proc. amer. math. soc., 95, 544-546, (1985) · Zbl 0596.30048
[15] Tse, K.-F., Nontangential interpolating sequences and interpolation by normal functions, Proc. amer. math. soc., 29, 351-354, (1971) · Zbl 0221.30036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.