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Sharp starlikeness conditions for analytic functions with bounded derivative

Part of: Geometric function theory

Published online by Cambridge University Press:  09 April 2009

Frode Rønning ,
Stephan Ruscheweyh  and
Nikolas Samaris
Frode Rønning
Affiliation:
Sør Trøndelag College School of Teacher EducationN-7004 TrondheimNorway e-mail: frode.ronning@alu.hist.no
Stephan Ruscheweyh
Affiliation:
Mathematisches Institut Universität WürzburgD-97074 WürzburgGermany e-mail: ruscheweyh@mathematik.uni-wuerzburg.de
Nikolas Samaris
Affiliation:
Department of Mathematics University of PatrasGR-26500 PatrasGreece e-mail: samaris@math.upatras.gr
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Abstract

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We develop sharp conditions for various types of starlikeness for functions analytic in the unit disk with bounded derivatives. We also describe the precise range {zf′(z)/f(z): z ∈ D, f ∈ }, where f ∈ means f (0) = 0, f′(0) = 1, and |f′(z) - 1 |< ≦ λ in the unit disc D, and draw some cnoslusions from that.

MSC classification

Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 3 , December 2000 , pp. 303 - 315
Copyright
Copyright © Australian Mathematical Society 2000

References

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