SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS
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Date
2011
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SUOMALAINEN TIEDEAKATEMIA
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 parallel to f parallel to <= 2. The inequality in sharper form leads to the conclusion that no convex mapping with parallel to f parallel to = 2 can map onto a quasidisk. In particular, every bounded convex mapping has Schwarzian norm parallel to f parallel to < 2. The analysis involves a structural formula for the pre-Schwarzian of a convex mapping, which is studied in further detail.
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Keywords
Convex mapping, Schwarzian derivative, Schwarzian norm, univalence, Schwarz lemma, Schwarz-Christoffel formula, quasidisk, John domain