Abstract
In the present paper, we introduce a family of univalent harmonic mappings, which map the unit disk onto domains convex in the direction of the imaginary axis. We find conditions for the linear combinations of mappings from this family to be univalent and convex in the direction of the imaginary axis. Linear combinations of functions from this family and harmonic mappings obtained by shearing of analytic vertical strip mappings are also studied.
Similar content being viewed by others
References
Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Suom. Tiedeakat. Ann. Sci. Acad. Finl. Ser. A 1: Math. 9, 3–25 (1984)
Dorff, M.: Anamorphosis, Mapping Problems and Harmonic Univalent Function, Explorations in Complex Analysis, pp. 197–269. Mathematical Association of America, Washington, DC (2012)
Hengartner, W., Schober, G.: On schlicht mappings to domains convex in one direction. Comment. Math. Helv. 45, 303–314 (1970)
Michalski, A.: Harmonic univalent functions convex in orthogonal direction. Ann. Univ. Mariae Curie-Sklodowska Lub. Polon. LX, 43–56 (2006)
Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials, London Mathematical Society Monographs New Series 26. Oxford University Press, Oxford (2002)
Wang, Z.-G., Liu, Z.-H., Li, Y.-C.: On the linear combinations of harmonic univalent mappings. J. Math. Anal. Appl. 400(2), 452–459 (2013)
Acknowledgments
The authors wish to thank the anonymous referees for their useful comments which helped to improve the manuscript in the present form. First author is also thankful to the Council of Scientific and Industrial Research, New Delhi, for financial support vide Grant No. 09/797/0006/2010 EMR-1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Lee See Keong.
Rights and permissions
About this article
Cite this article
Kumar, R., Gupta, S. & Singh, S. Linear Combinations of Univalent Harmonic Mappings Convex in the Direction of the Imaginary Axis. Bull. Malays. Math. Sci. Soc. 39, 751–763 (2016). https://doi.org/10.1007/s40840-015-0190-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-015-0190-5