Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T01:31:59.505Z Has data issue: false hasContentIssue false
  • English
  • Français

A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM II

Part of: Geometric function theory Two-dimensional theory

Published online by Cambridge University Press:  02 March 2017

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 [email protected]
ANDRZEJ M. MICHALSKI*
Affiliation:
Department of Complex Analysis, The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland email [email protected]
*
✉[email protected]
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.

We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

Keywords

harmonic mappingsimply connected setconvexity in the horizontal direction

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 95 , Issue 3 , June 2017 , pp. 457 - 466
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Clunie, J. G. and Sheil-Small, T., ‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. 9 (1984), 3–25.Google Scholar
Duren, P. L., Harmonic Mappings in the Plane, Cambridge Tracts in Math., 156 (Cambridge University Press, Cambridge, 2004).Google Scholar
Dorff, M., Nowak, M. and Wołoszkiewicz, M., ‘Harmonic mappings onto parallel slit domains’, Ann. Polon. Math. 101(2) (2011), 149–162.Google 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.Google Scholar
Ganczar, A. and Widomski, J., ‘Univalent harmonic mappings into two-slit domains’, J. Aust. Math. Soc. 88(1) (2010), 61–73.Google 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.Google 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. 59 (2009), 33–44; 58.Google Scholar
Livingston, A. E., ‘Univalent harmonic mappings’, Ann. Polon. Math. 57(1) (1992), 57–70.Google Scholar
Michalska, M. and Michalski, A. M., ‘A generalisation of the Clunie–Sheil-Small theorem’, Bull. Aust. Math. Soc. 96(1) (2016), 92–100.Google 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 Sairam Kaliraj, A., ‘On the coefficient conjecture of Clunie and Sheil-Small on univalent harmonic mappings’, Proc. Indian Acad. Sci. Math. 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.Google Scholar