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STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

Part of: Classical measure theory Miscellaneous topics of analysis in the complex domain Geometric function theory

Published online by Cambridge University Press:  05 October 2020

PENG-FEI ZHANG  and
XIN-HAN DONG
PENG-FEI ZHANG
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China e-mail: [email protected]
XIN-HAN DONG*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China
*
e-mail: [email protected]
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Abstract

For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

Keywords

starlikenessconvexityunivalenceCauchy transformregular polygons

MSC classification

Primary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions
Secondary: 28A80: Fractals 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 291 - 302
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This research is supported in part by the NNSF of China (No. 11831007).

References

Bell, S. R., The Cauchy Transform, Potential Theory and Conformal Mapping, 2nd edn (Chapman and Hall/CRC, Boca Raton, FL, 2016).Google Scholar
Dong, X. H. and Lau, K. S., ‘Cauchy transforms of self-similar measures: the Laurent coefficients’, J. Funct. Anal. 202(1) (2003), 67–97.CrossRefGoogle Scholar
Dong, X. H. and Lau, K. S., ‘An integral related to the Cauchy transform on the Sierpinski gasket’, Exp. Math. 13(4) (2004), 415–419.CrossRefGoogle Scholar
Dong, X. H. and Lau, K. S., ‘Cantor boundary behavior of analytic functions’, in: Recent Developments in Fractals and Related Fields, eds. Barrel, J. and Seuret, S. (Birkhäuser, Boston, MA, 2010), 283–294.CrossRefGoogle Scholar
Dong, X. H., Lau, K. S. and Liu, J. C., ‘Cantor boundary behavior of analytic functions’, Adv. Math. 232 (2013), 543–570.CrossRefGoogle Scholar
Dong, X. H., Lau, K. S. and Wu, H. H., ‘Cauchy transform of self-similar measures: Starlikeness and univalence’, Trans. Amer. Math. Soc. 369(7) (2017), 4817–4842.CrossRefGoogle Scholar
Dong, X. H., Wang, Y. and Zhang, P. F., ‘The starlikeness of Cauchy transform on square’, Comput. Methods Funct. Theory. 19(2) (2019), 341–351.CrossRefGoogle Scholar
Duren, P., Univalent Functions (Springer-Verlag, New York, 1983).Google Scholar
Garnett, J., Analytic Capacity and Measure, Lecture Notes in Mathematics, 297 (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
Hille, E., ‘Remarks on a paper by Zeev Nehari’, Bull. Amer. Math. Soc. 55 (1949), 552–553.CrossRefGoogle Scholar
Kress, R., Linear Integral Equations, 3rd edn (Springer, New York, 2014).CrossRefGoogle Scholar
Liu, J. C., Dong, X. H. and Peng, S. M., ‘A note on Cantor boundary behavior’, J. Math. Anal. Appl. 408(2) (2013), 795–801.CrossRefGoogle Scholar
Lund, J. P., Strichartz, R. and Vinson, J., ‘Cauchy transforms of self-similar measures’, Exp. Math. 7(3) (1998), 177–190.CrossRefGoogle Scholar
Nehari, Z., ‘The Schwarzian derivative and schlicht functions’, Bull. Amer. Math. Soc. 55 (1949), 545–551.CrossRefGoogle Scholar
Nehari, Z., ‘Some criteria of univalence’, Proc. Amer. Math. Soc. 5 (1954), 700–704.CrossRefGoogle Scholar
Zhang, P. F. and Dong, X. H., ‘Starlikeness and convexity of Cauchy transform on equilateral triangle’, Complex Var. Elliptic Equ. 65(9) (2020), 1590–1600.CrossRefGoogle Scholar