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ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS

Part of: Function theory on the disc Differential equations in the complex domain

Published online by Cambridge University Press:  17 December 2014

JANNE HEITTOKANGAS  and
ATTE REIJONEN
JANNE HEITTOKANGAS
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland e-mails: janne.heittokangas@uef.fi, atte.reijonen@uef.fi
ATTE REIJONEN
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland e-mails: janne.heittokangas@uef.fi, atte.reijonen@uef.fi
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Abstract

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If A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

MSC classification

Primary: 34M10: Oscillation, growth of solutions
Secondary: 30J10: Blaschke products
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 543 - 554
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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