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Certain classes of univalent functions with negative coefficients II

Published online by Cambridge University Press:  17 April 2009

V.P. Gupta  and
P.K. Jain
V.P. Gupta
Affiliation:
Department of Mathematics, M.M.H. College, University of Meerut, Ghaziabad, Uttar Pradesh, India;
P.K. Jain
Affiliation:
Department of Mathematics, University of Delhi, Delhi, India.
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Abstract

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Let P*(α, β) denote the class of functions

analytic and univalent in |z| < 1 for which

where α є [0, 1), β є (0, 1].

Sharp results concerning coefficients, distortion theorem and radius of convexity for the class P*(α, β) are determined. A comparable theorem for the classes C*(α, β) and P*(α, β) is also obtained. Furthermore, it is shown that the class P*(α, ß) is closed under ‘arithmetic mean’ and ‘convex linear combinations’.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 3 , December 1976 , pp. 467 - 473
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Gupta, V.P. and Jain, P.K., “Certain classes of univalent functions with negative coefficients”, Bull. Austral. Math. Soc. 14 (1976), 409–416.Google Scholar
[2]Schild, A., “On a class of functions schlicht in the unit circle”, Proc. Amer. Math. Soc. 5 (1954), 115–120.CrossRefGoogle Scholar
[3]Silverman, Herb, “Univalent functions with negative coefficients”, Proc. Amer. Math. Soc. 51 (1975), 109–116.Google Scholar