Convex combinations of harmonic shears of slit mappings

Shi, L.; Wang, Z.-G.; Rasila, A.; Sun, Y.

Convex combinations of harmonic shears of slit mappings

Document Type : Research Paper

Authors

¹ School of Mathematics and Statistics‎, ‎Anyang Normal University‎, ‎Anyang 455002‎, ‎Henan‎, ‎P‎.‎R‎. ‎China.

² School of Mathematics and Computing Science‎, ‎Hunan First Normal University‎, ‎Changsha 410205‎, ‎Hunan‎, ‎P‎.‎R‎. ‎China.

³ Department of Mathematics and Systems Analysis‎, ‎Aalto University‎, ‎P‎.‎O‎. ‎Box 11100‎, ‎FI-00076 Aalto‎, ‎Finland.

⁴ School of Science‎, ‎Hunan Institute of Engineering‎, ‎Xiangtan 411104‎, ‎Hunan‎, ‎P‎.‎R‎. ‎China.

Abstract

‎In this paper‎, ‎we study the convex combinations of harmonic mappings obtained by shearing a class of slit conformal mappings‎. ‎Sufficient conditions for the convex combinations of harmonic mappings of this family to be univalent and convex in the horizontal direction are derived‎. ‎Several examples of univalent harmonic mappings constructed by using these methods are presented to illustrate potential applications of the main results.

Keywords

Main Subjects

30-XX Functions of a complex variable

References

D.M. Campbell, A survey of properties of the convex combination of univalent functions, Rocky Mountain J. Math. 5 (1975), no. 4, 475--492.

J. Clunie and T. Sheil-Smail, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9 (1984) 3--25.

A. Cohn, Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z. 14 (1922), no. 1, 110--148.

M. Dorff, Anamorphosis, mapping problems, and harmonic univalent functions, Explorations in Complex Analysis, pp. 197--269, Classr. Res. Mater. Ser. Math. Assoc. America, Washington, DC, 2012.

M. Dorff, M. Nowak and M. Woloszkiewicz, Harmonic mappings onto parallel slit domains, Ann. Polon. Math. 101 (2011), no. 2, 149--162.

R. Kumar, S. Gupta and S. Singh, Linear combinations of univalent harmonic mappings convex in the direction of the imaginary axis, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 2, 751--763.

T.H. Macgregor, The univalence of a linear combination of convex mappings, J. Lond. Math. Soc. 44 (1969) 210--212.

Z. Nehari, Conformal Mapping, Reprinting of the 1952 edition, Dover Publications, New York, 1952.

S. Ponnusamy, T. Quach and A. Rasila, Harmonic shears of slit and polygonal mappings, Appl. Math. Comput. 233 (2014) 588--598.

Y. Sun, A. Rasila and Y.P. Jiang, Linear combinations of harmonic quasiconformal mappings convex in one direction, Kodai Math. J. 39 (2016), no. 2, 366--377.

S.Y. Trimble, The convex sum of convex functions, Math. Z. 109 (1969) 112--114.

Z.G. Wang, Z.H. Liu and Y.C. Li, On the linear combinations of harmonic univalent mappings, J. Math. Anal. Appl. 400 (2013), no. 2, 452--459.

Z.G. Wang, L. Shi and Y.P. Jiang, Construction of harmonic univalent mappings convex in one direction, Sci. Sin. Math. 40 (2014) 139--150.