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Univalent harmonic mappings convex in one direction

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Abstract

In this paper, we present a criterion for a harmonic function to be convex in one direction. Also, we discuss the class of harmonic functions starlike in one direction in the unit disk \({\mathbb D}\) and obtain a method to construct univalent harmonic functions convex in one direction. Although the converse of classical Alexander’s theorem for harmonic functions was proved to be false, we obtain a version of converse of it under a suitable additional condition.

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Acknowledgments

The second author thanks Council of Scientific and Industrial Research (CSIR), India, for providing financial support in the form of a Senior Research Fellowship to carry out this research. The authors thank the referee for valuable comments.

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Authors and Affiliations

  1. Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and security), MGR Knowledge City, CIT Campus, Taramani, Chennai, 600 113, India

    S. Ponnusamy

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    A. Sairam Kaliraj

Authors
  1. S. Ponnusamy
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  2. A. Sairam Kaliraj
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Correspondence to S. Ponnusamy.

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This author is on leave from the Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India.

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Ponnusamy, S., Kaliraj, A.S. Univalent harmonic mappings convex in one direction. Anal.Math.Phys. 4, 221–236 (2014). https://doi.org/10.1007/s13324-013-0066-5

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  • DOI: https://doi.org/10.1007/s13324-013-0066-5

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