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On analytic functions with reference to an integral operator

Published online by Cambridge University Press:  17 April 2009

R. Parvatham  and
T.N. Shanmugam
R. Parvatham
Affiliation:
The Ramanujam Institute for Advanced Study in Mathematics, University of Madras, Madras 600 005, India;
T.N. Shanmugam
Affiliation:
Department of Mathematics, Anna University, Madras 600 025, India.
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Abstract

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Let E = {z: |z| < 1} and let H = {w : regular in E, w(0) = 0, |w(z)| < l, z ∈ E}.

Let P(A, B) denote the class of functions in E which can be put in the form (1 + Aw(z))/(1 + Bw(z)), −1 ≤ A < B ≤ 1, w(z) ∈ H. Let S*(A, B) denote the class of functions f(z) of the form such that zf′(z)/f(z) ∈ P(A, B). If f(z) ∈ S*(A, B) and g(z) ∈ S*(C, D) then, in this paper the radius of starlikeness of order β (β ∈ [0, 1]) of the following integral operator

is determined. Conversely, a sharp estimate is obtained for the radius of starlikeness of the class of functions

where g(z) and F(z) belong to the class S*(A, B).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 28 , Issue 2 , October 1983 , pp. 207 - 215
Copyright
Copyright © Australian Mathematical Society 1983

References

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