Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 36, 2011, 449-460

SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS

Martin Chuaqui, Peter Duren and Brad Osgood

P. Universidad Católica de Chile, Facultad de Matemáticas
Casilla 306, Santiago 22, Chile; mchuaqui 'at' mat.puc.cl

University of Michigan, Department of Mathematics
Ann Arbor, Michigan 48109-1043, U.S.A.; duren 'at' umich.edu

Stanford University, Department of Electrical Engineering
Stanford, California 94305, U.S.A.; osgood 'at' ee.stanford.edu

Abstract. A simple proof is given for Nehari's theorem that an analytic function f which maps the unit disk onto a convex region has Schwarzian norm ||Sf|| \leq 2. The inequality in sharper form leads to the conclusion that no convex mapping with ||Sf|| = 2 can map onto a quasidisk. In particular, every bounded convex mapping has Schwarzian norm ||Sf|| < 2. The analysis involves a structural formula for the pre-Schwarzian of a convex mapping, which is studied in further detail.

2010 Mathematics Subject Classification: Primary 30C45; Secondary 30C80, 30C62.

Key words: Convex mapping, Schwarzian derivative, Schwarzian norm, univalence, Schwarz lemma, Schwarz-Christoffel formula, quasidisk, John domain.

Reference to this article: M. Chuaqui, P. Duren and B. Osgood: Schwarzian derivatives of convex mappings. Ann. Acad. Sci. Fenn. Math. 36 (2011), 449-460.

Full document as PDF file

doi:10.5186/aasfm.2011.3628

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