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.
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The second author thanks Council of Scientific and Industrial Research (CSIR), India, for providing financial support in the form Junior Research Fellowship to carry out this research.
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Ponnusamy, S., Kaliraj, A.S. On Harmonic Close-To-Convex Functions. Comput. Methods Funct. Theory 12, 669–685 (2012). https://doi.org/10.1007/BF03321849
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DOI: https://doi.org/10.1007/BF03321849
Keywords
- Coefficient inequality
- univalence
- close-to-convex
- univalent harmonic functions
- Gaussian hypergeometric functions