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A Survey on Blaschke-Oscillatory Differential Equations, with Updates

  • Chapter
Blaschke Products and Their Applications

Part of the book series: Fields Institute Communications ((FIC,volume 65))

Abstract

In the celebrated 1949 paper due to Nehari, necessary and sufficient conditions are given for a locally univalent meromorphic function to be univalent in the unit disc \(\mathbb{D}\). The proof involves a second order differential equation of the form

$$ f''+A(z)f=0, $$
(†)

where A(z) is analytic in \(\mathbb{D}\). As an immediate consequence of the proof, it follows that if |A(z)|≤1/(1−|z|2)2 for every \(z\in\mathbb{D}\), then any non-trivial solution of (†) has at most one zero in \(\mathbb{D}\).

Since 1949 a number of papers provide with different types of growth conditions on the coefficient A(z) such that the solutions of (†) have at most finitely many zeros in \(\mathbb{D}\). If there exists at least one solution with infinitely many zeros in \(\mathbb{D}\), then (†) is oscillatory. If the zeros still satisfy the classical Blaschke condition, then (†) is called Blaschke-oscillatory. This concept was introduced by the author in 2005, but the topic was considered by Hartman and Wintner already in 1955 (Trans. Am. Math. Soc. 78:492–500). This semi-survey paper provides with a collection of results and tools dealing with Blaschke-oscillatory equations.

As for results, necessary and sufficient conditions are given, and notable effort has been put in dealing with prescribed zero sequences satisfying the Blaschke condition. The concept of Blaschke-oscillation also extends to differential equations of arbitrary order. Many of the results given in this paper have been published earlier in a weaker form. All questions regarding the zeros of solutions can be rephrased for the critical points of solutions. This gives rise to a new concept called Blaschke-critical equations. To intrigue the reader, several open problems are pointed out in the text.

Some classical tools and closely related topics that are often related to the finite oscillation case include the Schwarzian derivative, properties of univalent functions, Green’s identity, conformal mappings, and a certain Hardy-Littlewood inequality. The Blaschke-oscillatory case also makes use of interpolation theory, various growth estimates for logarithmic derivatives of Blaschke products, Bank-Laine functions and recently updated Wiman-Valiron theory.

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Notes

  1. 1.

    This part of the proof can be done without appealing to the second fundamental theorem, see Sect. 4.2

  2. 2.

    Since W′(f,g)(z)≡0 by (1), we have W(f,g)(z)≡c. If c=0, then f′/f=g′/g, and hence f and g are linearly dependent.

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  1. Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101, Joensuu, Finland

    Janne Heittokangas

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  1. , Département de mathématiques et de stati, Université Laval, avenue de la Médecine 1045, Quebec, G1K 7P4, Québec, Canada

    Javad Mashreghi

  2. , Institut Camille Jordan, Université Claude Bernard Lyon 1, avenue du 11 novembre 1918 43, Villeurbanne cedex, 69622, France

    Emmanuel Fricain

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Heittokangas, J. (2013). A Survey on Blaschke-Oscillatory Differential Equations, with Updates. In: Mashreghi, J., Fricain, E. (eds) Blaschke Products and Their Applications. Fields Institute Communications, vol 65. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5341-3_3

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