Annales Academić Scientiarum Fennicć
Mathematica
Volumen 33, 2008, 561-584

VALENCE AND OSCILLATION OF FUNCTIONS IN THE UNIT DISK

Martin Chuaqui and Dennis Stowe

Pontificia Universidad Católica de Chile, Facultad de Matemáticas
Santiago, Chile; mchuaqui 'at' mat.puc.cl

Idaho State University, Mathematics Department
Pocatello ID 83209, U.S.A.; stowdenn 'at' isu.edu

Abstract. We investigate the number of times that nontrivial solutions of equations u'' + p(z)u = 0 in the unit disk can vanish - or, equivalently, the number of times that solutions of S(f) = 2p(z) can attain their values - given a restriction |p(z)| \leq b(|z|). We establish a bound for that number when b satisfies a Nehari-type condition, identify perturbations of the condition that allow the number to be infinite, and compare those results with their analogs for real equations \varphi'' + q(t)\varphi = 0 in (-1,1).

2000 Mathematics Subject Classification: Primary 34M10, 34C10, 30C55.

Key words: Valence, oscillation, Schwarzian derivative.

Reference to this article: M. Chuaqui and D. Stowe: Valence and oscillation of functions in the unit disk. Ann. Acad. Sci. Fenn. Math. 33 (2008), 561-584.

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