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Admissible solutions of the Schwarzian differential equation

Part of: Entire and meromorphic functions, and related topics Qualitative theory

Published online by Cambridge University Press:  09 April 2009

Katsuya Ishizaki
Katsuya Ishizaki
Affiliation:
Department of Mathematics Tokyo National College of Technology1220 -2 Kunugida-cho Hachioji Tokyo 193, Japan
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Abstract

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Let R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation possesses an admissible solution, then , where αj, are distinct complex constants. In particular, when R(z, w) is independent of z, it is shown that if (*) possesses an admissible solution w(z), then by some Möbius transformation u = (aw + b) / (cw + d) (ad – bc ≠ 0), the equation can be reduced to one of the following forms: where τj (j = 1, … 4) are distinct constants, and σj (j = 1, … 4) are constants, not necessarily distinct.

Keywords

admissible solutionSchwarzianNevanlinna theory

MSC classification

Secondary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 2 , April 1991 , pp. 258 - 278
Copyright
Copyright © Australian Mathematical Society 1991

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