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REMARKS ON THE UNIVALENCE CRITERION OF PASCU AND PASCU

Part of: Geometric function theory

Published online by Cambridge University Press:  22 August 2013

VAIDHYANATHAN BHARANEDHAR  and
SAMINATHAN PONNUSAMY
VAIDHYANATHAN BHARANEDHAR
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India email [email protected]
SAMINATHAN PONNUSAMY*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India
*
✉[email protected],[email protected]
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Abstract

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We consider a recent work of Pascu and Pascu [‘Neighbourhoods of univalent functions’, Bull. Aust. Math. Soc. 83(2) (2011), 210–219] and rectify an error that appears in their work. In addition, we study certain analogous results for sense-preserving harmonic mappings in the unit disc $\vert z\vert \lt 1$. As a corollary to this result, we derive a coefficient condition for a sense-preserving harmonic mapping to be univalent in $\vert z\vert \lt 1$.

Keywords

analyticunivalentconvexclose-to-convex and harmonic mappings

MSC classification

Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C50: Coefficient problems for univalent and multivalent functions 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 2 , April 2014 , pp. 210 - 216
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory, 2nd edn, Graduate Texts in Mathematics, 137 (Springer, New York, 2001).CrossRefGoogle Scholar
Clunie, J. G. and Sheil-Small, T., ‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A.I. 9 (1984), 3–25.Google Scholar
Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259 (Springer, New York, 1983).Google Scholar
Duren, P., Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Lewy, H., ‘On the nonvanishing of the Jacobian in certain one-to-one mappings’, Bull. Amer. Math. Soc. 42 (1936), 689–692.Google Scholar
Pascu, M. N. and Pascu, N. R., ‘Neighbourhoods of univalent functions’, Bull. Aust. Math. Soc. 83 (2) (2011), 210–219.Google Scholar
Ponnusamy, S., Yamamoto, H. and Yanagihara, H., ‘Variability regions for certain families of harmonic univalent mappings’, Complex Var. Elliptic Equ. 58 (1) (2013), 23–34.CrossRefGoogle Scholar