Convolution properties of the harmonic Koebe function and its connection with 2-starlike mappings

Abstract

In 1984, Clunie and Sheil-Small constructed the harmonic Koebe function expecting it to play the role of extremal function for extremal problems over the class of sense-preserving harmonic univalent functions suitably normalized in the open unit disc. In this paper, we investigate the convolution properties of . In addition, we discuss the geometric properties of -starlike functions defined using the Sălăgean differential operator. Given a -starlike function , the product is shown to be univalent and convex in the direction of the real axis.

Keywords:

• univalent harmonic mappings
• sense-preserving
• starlike
• convex
• close-to-convex
• convex in one direction
• convolution
• Sălăgean operator

AMS Subject Classifications:

• Primary: 31A05
• Secondary: 30C45

Acknowledgements

The research work presented here is supported by research fellowship from Council of Scientific and Industrial Research (CSIR), New Delhi and a grant from University of Delhi, Delhi. The authors are thankful to the referees for their useful comments.