Annales Academiĉ Scientiarum Fennicĉ

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*) = 2*p*(*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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