Opuscula Math. 33, no. 2 (2013), 273-281
http://dx.doi.org/10.7494/OpMath.2013.33.2.273

 
Opuscula Mathematica

A unified representation of some starlike and convex harmonic functions with negative coefficients

R. M. El-Ashwah
M. K. Aouf
A. A. M. Hassan
A. H. Hassan

Abstract. In this paper we introduce a unified representation of starlike and convex harmonic functions with negative coefficients, related to uniformly starlike and uniformly convex analytic functions. We obtain extreme points, distortion bounds, convolution conditions and convex combinations for this family.

Keywords: harmonic, analytic, univalent, sense-preserving, extreme points.

Mathematics Subject Classification: 30C45.

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  • R. M. El-Ashwah
  • Damietta University, Faculty of Science, Department of Mathematics, New Damietta 34517, Egypt
  • M. K. Aouf
  • Mansoura University, Faculty of Science, Department of Mathematics, Mansoura 33516, Egypt
  • A. A. M. Hassan
  • Zagazig University, Faculty of Science, Department of Mathematics, Zagazig 44519, Egypt
  • A. H. Hassan
  • Zagazig University, Faculty of Science, Department of Mathematics, Zagazig 44519, Egypt
  • Received: 2012-06-23.
  • Revised: 2012-10-29.
  • Accepted: 2012-10-29.
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Cite this article as:
R. M. El-Ashwah, M. K. Aouf, A. A. M. Hassan, A. H. Hassan, A unified representation of some starlike and convex harmonic functions with negative coefficients, Opuscula Math. 33, no. 2 (2013), 273-281, http://dx.doi.org/10.7494/OpMath.2013.33.2.273

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