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An integral mean inequality for starlike functions

Part of: Geometric function theory

Published online by Cambridge University Press:  26 February 2010

J. B. Twomey
J. B. Twomey
Affiliation:
Department of Mathematics, University College, Cork, Ireland.
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Extract

A function

analytic and univalent in U = {z: |z| < 1} is said to be starlike there, if f(U) is f starshaped with respect to the origin, that is, if w ε f(U) implies tw ε f(U) for 0 ≤t ≤ 1. We denote by S* the class of all such functions. The Koebe function; k(z) = z(l – z)-2, z ε U, maps U onto the complex plane minus a slit along the I negative real axis from - ¼ to ∞, and thus belongs to the class S*. Recently Leung [4] has shown that, if

then, for f ε S*,

for every p > 0.

MSC classification

Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Mathematika , Volume 28 , Issue 1 , June 1981 , pp. 88 - 98
Copyright
Copyright © University College London 1981

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References

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