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Linear Combinations of Univalent Harmonic Mappings Convex in the Direction of the Imaginary Axis

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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.

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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.

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Authors and Affiliations

  1. DAV University, Jalandhar, Punjab, 144001, India

    Raj Kumar

  2. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, 148106, India

    Sushma Gupta & Sukhjit Singh

Authors
  1. Raj Kumar
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  2. Sushma Gupta
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  3. Sukhjit Singh
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Correspondence to Raj Kumar.

Additional information

Communicated by Lee See Keong.

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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

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  • DOI: https://doi.org/10.1007/s40840-015-0190-5

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