Ein Verzerrungssatz für schlichte Funktionen

Blatter, Christian

doi:10.1007/bf02566106

Cited by 18 publications

(8 citation statements)

References 2 publications

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“…We will also obtain a two-point distortion theorem that actually characterizes Nehari functions. This result can be viewed as an analogue of a theorem of Blatter that characterizes the set of all univalent functions in D [2].…”

Section: Introductionmentioning

confidence: 94%

Characteristic properties of Nehari functions

Chuaqui¹,

Pommerenke²

1999

Pacific J. Math.

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Let N be the set of all meromorphic functions f defined in the unit disc D that satisfy Nehari's univalence criterion (1 − |z| 2 ) 2 |Sf (z)| ≤ 2. In this paper we investigate certain properties of the class N . We obtain sharp estimates for the spherical distortion, and also a two-point distortion theorem that actually characterizes the set N . Finally, we study some aspects of the boundary behavior of Nehari functions, and obtain results that indicate how such maps can fail to map D onto a quasidisc.

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Section: Introductionmentioning

confidence: 94%

Characteristic properties of Nehari functions

Chuaqui¹,

Pommerenke²

1999

Pacific J. Math.

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“…Some years ago, Blatter [1] found a general two-point distortion theorem that is both necessary and sufficient for univalence and requires no normalization. His result is formulated in terms of the hyperbolic metric …”

Section: Introductionmentioning

confidence: 99%

Two-point distortion theorems for harmonic and pluriharmonic mappings

Duren

Hamada

Kohr

2011

Trans. Amer. Math. Soc.

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“…Another measure of distortion is the distance |f (z 1 ) − f (z 2 )| between the images of two arbitrary points in the disk. Some years ago, Blatter [3] gave a sharp lower bound for this distance in terms of the hyperbolic distance between z 1 and z 2 . More recently, Chuaqui and Pommerenke [10] found a sharp two-point distortion theorem for functions whose Schwarzian derivative satisfies Nehari's condition |Sf (z)| ≤ 2(1 − |z| 2 ) −2 .…”

Section: §1 Introductionmentioning

confidence: 99%

“…Under the same hypotheses it turns out that the harmonic lift f actually satisfies a two-point distortion condition. The inequality will involve the function F determined by a Nehari function p as in the formula (3). In order to state the result in most elegant form, it will be convenient to assume that the given Nehari function is extremal, as defined in Section 1.…”

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confidence: 99%

Two-point distortion theorems for harmonic mappings

Chuaqui¹,

Duren²,

Osgood³

2009

Illinois J. Math.

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In earlier work the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in R n are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form. §1. Introduction.The classical Koebe distortion theorem gives sharp bounds on the derivative of a normalized analytic univalent function. Another measure of distortion is the distance |f (z 1 ) − f (z 2 )| between the images of two arbitrary points in the disk. Some years ago, Blatter [3] gave a sharp lower bound for this distance in terms of the hyperbolic distance between z 1 and z 2 . More recently, Chuaqui and Pommerenke [10] found a sharp two-point distortion theorem for functions whose Schwarzian derivative satisfies Nehari's condition |Sf (z)| ≤ 2(1 − |z| 2 ) −2 . Their result may be viewed as a quantitative form of Nehari's univalence criterion. The main purpose of the present paper is to carry out a similar analysis for harmonic mappings, or rather for their canonical lifts to minimal surfaces. Along the way we obtain distortion theorems for curves in R n , thereby recasting an injectivity criterion of Chuaqui and Gevirtz [7] in quantitative form.An important tool throughout the paper is the classical Sturm comparison theorem for solutions of linear differential equations of second order. A good reference for this topic is the book of Birkhoff and Rota [2].

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Ein Verzerrungssatz für schlichte Funktionen

Cited by 18 publications

References 2 publications

Characteristic properties of Nehari functions

Characteristic properties of Nehari functions

Two-point distortion theorems for harmonic and pluriharmonic mappings

Two-point distortion theorems for harmonic mappings

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