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Computational Methods and Function Theory

, Volume 12, Issue 2, pp 669–685 | Cite as

On Harmonic Close-To-Convex Functions

  • Saminathan Ponnusamy
  • Anbareeswaran Sairam Kaliraj
Article

Abstract

In this paper, we study the family of sense-preserving complex-valued harmonic functions f that are normalized and close-to-convex on the open unit disk D. First we investigate the conditions for which f is close-to-convex on D. As a consequence, we derive a sufficient condition for f to be in this family. Using the condition, we establish sufficient conditions for f to be close-to-convex, in terms of the coefficients of the analytic and the co-analytic parts of f. Finally, we determine conditions on a, b such that \(f(z)=zF(a,b;a+b;z)+\overline{\alpha z^{2}F(a,b;a+b;z)}\) is harmonic close-to-convex (and hence univalent) in D, where F(a, b; c; z) denotes the Gaussian hypergeometric function. A similar result, and a number of interesting corollaries and examples of harmonic close-to-convex functions, are also obtained.

Keywords

Coefficient inequality univalence close-to-convex univalent harmonic functions Gaussian hypergeometric functions 

2000 MSC

30C45 

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Copyright information

© Heldermann  Verlag 2012

Authors and Affiliations

  • Saminathan Ponnusamy
    • 1
  • Anbareeswaran Sairam Kaliraj
    • 1
  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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