SCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS

被引：16

作者：

Chuaqui, Martin ^{[1
]}

Duren, Peter ^{[2
]}

Osgood, Brad ^{[3
]}

机构：

[1] Pontificia Univ Catolica Chile, Fac Matemat, Santiago 22, Chile

[2] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

[3] Stanford Univ, Dept Elect Engn, Stanford, CA 94305 USA

来源：

ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA | 2011年 / 36卷 / 02期

关键词：

Convex mapping; Schwarzian derivative; Schwarzian norm; univalence; Schwarz lemma; Schwarz-Christoffel formula; quasidisk; John domain;

D O I：

10.5186/aasfm.2011.3628

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页码：449 / 460

页数：12

共 15 条

[1]

Ahlfors L. V., 1962, P AM MATH SOC, V13, P975, DOI [DOI 10.2307/2034099, DOI 10.1090/S0002-9939-1962-0148896-1]

[2]

[Anonymous], 1992, GRUNDLEHREN MATH WIS

[3]

Bieberbach L., 2000, CONFORMAL MAPPING

[4] SHARP DISTORTION-THEOREMS ASSOCIATED WITH THE SCHWARZIAN DERIVATIVE [J].

CHUAQUI, M ;

OSGOOD, B .

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 1993, 48 :289-298

[5]

Chuaqui M, 1996, J REINE ANGEW MATH, V471, P77

[6] AHLFORS-WEILL EXTENSIONS OF CONFORMAL-MAPPINGS AND CRITICAL-POINTS OF THE POINCARE METRIC [J].

CHUAQUI, M ;

OSGOOD, B .

COMMENTARII MATHEMATICI HELVETICI, 1994, 69 (04) :659-668

[7]

Duren P. L., 1983, Univalent Functions

[8] ON THE NEHARI UNIVALENCE CRITERION AND QUASICIRCLES [J].

GEHRING, FW ;

POMMERENKE, C .

COMMENTARII MATHEMATICI HELVETICI, 1984, 59 (02) :226-242

[9]

Hummel J, 1957, PACIFIC J MATH, V7, P1381, DOI DOI 10.2140/PJM.1957.7.1381

[10]

Kim S-A., 1993, J. Anal, V1, P109

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