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A radius of convexity problem

Published online by Cambridge University Press:  17 April 2009

K.S. Padmanabhan  and
M.S. Ganesan
K.S. Padmanabhan
Affiliation:
Ramanujan Institute of Advanced Study in Mathematics, University of Madras, Chepauk, Madras 600005, Tamil Nadu, India;
M.S. Ganesan
Affiliation:
Department of Mathematics, AVVM Sri Pushpam College, Poondi, Thanjavur District, Tamil Nadu, India.
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Abstract

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The authors determine the sharp radius of convexity for functions analytic in the unit disc having power series representation of the form f(z) = z + an+1zn+1 + an+2zn+2 + … where an+1 is fixed and such that zf′(z)/f(x) = (1 + Aw(z))/(1 + Bw(z)), −1 ≤ B < 0 < A ≤ 1 where w(z) is an analytic function satisfying the conditions of Schwarz's lemma, in the case A + B ≥ 0. The estimate obtained is an improvement over the corresponding result obtained by Mogra and Juneja for functions analytic and starlike in the unit disc, with missing coefficients where the initial non-vanishing coefficient is fixed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 28 , Issue 3 , December 1983 , pp. 433 - 439
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Goluzin, G.M., “Geometric theory of functions of a complex variable”, Amer. Math. Soc. Transl. 26 (1969), 329–335.Google Scholar
[2]Gronwall, T.H., “On the distortion in conformal mapping when the second coefficient in the mapping function has an assigned value”, Proc. Nat. Acad. Sci. USA 6 (1920), 300–302.CrossRefGoogle ScholarPubMed
[3]Juneja, O.P. and Mogra, M.L., “A radius of convexity problem”, Bull. Austral. Math. Soc. 24 (1981), 381–388.Google Scholar
[4]McCarty, Carl P., “Two radius of convexity problems”, Proc. Amer. Math. Soc 37 (1979), 153–160.Google Scholar
[5]Shaffer, D.B., “On bounds for the derivative of analytic functions”, Proc. Amer. Math. Soc. 37 (1973), 517–520.CrossRefGoogle Scholar
[6]Shaffer, D.B., “Theorems for a special class of analytic functions”, Proc. Amer. Math. Soc. 39 (1973), 281–287.CrossRefGoogle Scholar
[7]Stankiewicz, Jan and Waniurski, Jozei, “Some classes of functions subordinate to linear transformation and their applications”, Ann. Univ. Mariae Curie-Skhodowska Sect. A 28 (1974), 85–93.Google Scholar