Document Type : Research Paper

Authors

1 Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India.

2 Department of Mathematics, Hans Raj Mahila Maha Vidyalya, Jalandhar-144001, Punjab, India.

3 Department of Mathematics, DAV University, Jalandhar-144001, Punjab, India.

4 School of Advanced Science,Vellore Institute of technology Deemed to be University, Vellore - 632014, India.

10.22130/scma.2021.132155.841

Abstract

We define a new subclass of univalent harmonic mappings using multiplier transformation and investigate various properties like necessary and sufficient conditions, extreme points, starlikeness,  radius of convexity. We prove that the class is closed under harmonic convolutions and convex combinations. Finally, we show that this class is invariant under Bernandi-Libera-Livingston integral for harmonic functions.

Keywords

[1] K. Al-Shaqsi, M. Darus and O. Abidemi, A New Subclass of Salagean-Type Harmonic Univalent Functions, Abstract and Appl. Ana., (2010), Article ID 821531, pp. 1-12.

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

[3] J.M. Jahangiri, Harmonic functions starlike in the unit disk, Journal of Mathematical Analysis and Applications, 235(2) (1999), pp. 470-477.

[4] J.M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, Salagean type harmonic functions, Southwest J. Pure Appl. Math., 2(2002), pp. 77-82.

[5] R. Kumar, S. Gupta and S. Singh, A class of univalent harmonic functions defined by multiplier transformation, Rev. Roumaine Math. Pures Appl., 57 (2012), pp. 371-382.

[6] H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc., 42 (1936), pp. 689-692.

[7] T.O. Opoola, On a new subclass of univalent functions, Mathematica, 36(59) (1994), pp. 195-200.

[8] G.S. Salagean, Subclasses of univalent functions, Lecture Notes in Math., Springer - Verlag, Heidlberg, 1013 (1983), pp. 362-372.

[9] H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl., 220 (1998), pp. 283-289.

[10] E. Yasar and S. Yalcin, Harmonic univalent functions starlike or convex of complex order, Tamsui Oxford Journal of Information and Mathematical Sciences, 27(3) (2011), pp. 269-277.