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The Starlikeness of Cauchy Transform on Square

The Starlikeness of Cauchy Transform on Square Suppose that K is the square with vertexes $$\{1, i, -1, -i\}$$ { 1 , i , - 1 , - i } and $$\mu =\frac{1}{2}{\mathcal {L}}^2$$ μ = 1 2 L 2 is the normalized two-dimensional Lebesgue measure on K, let F(z) be the Cauchy transform of $$\mu $$ μ . Dong et al. (Trans Am Math Soc 369:4817–4842, 2017) proved that F(z) is univalent in $$\widehat{\mathbb {C}} {\setminus } K$$ C ^ \ K . In this paper, we show that F(z) is starlike in $$\widehat{\mathbb {C}} {\setminus } K$$ C ^ \ K , but not convex in $$\widehat{\mathbb {C}} {\setminus } K$$ C ^ \ K . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

The Starlikeness of Cauchy Transform on Square

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00267-y
Publisher site
See Article on Publisher Site

Abstract

Suppose that K is the square with vertexes $$\{1, i, -1, -i\}$$ { 1 , i , - 1 , - i } and $$\mu =\frac{1}{2}{\mathcal {L}}^2$$ μ = 1 2 L 2 is the normalized two-dimensional Lebesgue measure on K, let F(z) be the Cauchy transform of $$\mu $$ μ . Dong et al. (Trans Am Math Soc 369:4817–4842, 2017) proved that F(z) is univalent in $$\widehat{\mathbb {C}} {\setminus } K$$ C ^ \ K . In this paper, we show that F(z) is starlike in $$\widehat{\mathbb {C}} {\setminus } K$$ C ^ \ K , but not convex in $$\widehat{\mathbb {C}} {\setminus } K$$ C ^ \ K .

Journal

Computational Methods and Function TheorySpringer Journals

Published: Apr 29, 2019

References