Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T06:08:40.130Z Has data issue: false hasContentIssue false
  • English
  • Français

A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM

Part of: Two-dimensional theory Geometric function theory

Published online by Cambridge University Press:  17 February 2016

MAŁGORZATA MICHALSKA  and
ANDRZEJ M. MICHALSKI
MAŁGORZATA MICHALSKA
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland email malgorzata.michalska@poczta.umcs.lublin.pl
ANDRZEJ M. MICHALSKI*
Affiliation:
Department of Complex Analysis, The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland email amichal@kul.lublin.pl
*
✉amichal@kul.lublin.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction.

Keywords

harmonic mappingsconvex in one directionshear construction

MSC classification

Primary: 31A05: Harmonic, subharmonic, superharmonic functions
Secondary: 30C55: General theory of univalent and multivalent functions 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 92 - 100
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Clunie, J. G. and Sheil-Small, T., ‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984), 3–25.Google Scholar
Duren, P. L., Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Dorff, M., Nowak, M. and Woşoszkiewicz, M., ‘Harmonic mappings onto parallel slit domains’, Ann. Polon. Math. 101(2) (2011), 149–162.CrossRefGoogle Scholar
Dorff, M. and Szynal, J., ‘Harmonic shears of elliptic integrals’, Rocky Mountain J. Math. 35(2) (2005), 485–499.CrossRefGoogle Scholar
Driver, K. and Duren, P., ‘Harmonic shears of regular polygons by hypergeometric functions’, J. Math. Anal. Appl. 239(1) (1999), 72–84.CrossRefGoogle Scholar
Ganczar, A. and Widomski, J., ‘Univalent harmonic mappings into two-slit domains’, J. Aust. Math. Soc. 88(1) (2010), 61–73.CrossRefGoogle Scholar
Greiner, P., ‘Geometric properties of harmonic shears’, Comput. Methods Funct. Theory 4(1) (2004), 77–96.Google Scholar
Grigorian, A. and Szapiel, W., ‘Two-slit harmonic mappings’, Ann. Univ. Mariae Curie-Skłodowska Sect. A 49 (1995), 59–84.Google Scholar
Hengartner, W. and Schober, G., ‘Univalent harmonic functions’, Trans. Amer. Math. Soc. 299(1) (1987), 1–31.CrossRefGoogle Scholar
Klimek-Smȩt, D. and Michalski, A., ‘Univalent harmonic functions generated by conformal mappings onto regular polygons’, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 58 (2009), 33–44.Google Scholar
Kuratowski, K., Introduction to Set Theory and Topology (Pergamon Press, Oxford–London–New York–Paris, 1961).Google Scholar
Livingston, A. E., ‘Univalent harmonic mappings’, Ann. Polon. Math. 57(1) (1992), 57–70.CrossRefGoogle Scholar
Lyzzaik, A. K. and Stephenson, K., ‘The structure of open continuous mappings having two valences’, Trans. Amer. Math. Soc. 327(2) (1991), 525–566.CrossRefGoogle Scholar
Ortel, M. and Smith, W., ‘A covering theorem for continuous locally univalent maps of the plane’, Bull. Lond. Math. Soc. 18(4) (1986), 359–363.CrossRefGoogle Scholar
Ponnusamy, S., Quach, T. and Rasila, A., ‘Harmonic shears of slit and polygonal mappings’, Appl. Math. Comput. 233 (2014), 588–598.Google Scholar
Ponnusamy, S. and Qiao, J., ‘Classification of univalent harmonic mappings on the unit disk with half-integer coefficients’, J. Aust. Math. Soc. 98(2) (2015), 257–280.Google Scholar
Ponnusamy, S. and Sairam Kaliraj, A., ‘On the coefficient conjecture of Clunie and Sheil-Small on univalent harmonic mappings’, Proc. Indian Acad. Sci. 125(3) (2015), 277–290.Google Scholar
Starkov, V. V., ‘Univalence of harmonic functions, problem of Ponnusamy and Sairam, and constructions of univalent polynomials’, Probl. Anal. Issues Anal. 3(21)(2) (2014), 59–73.CrossRefGoogle Scholar