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ON BRANNAN'S COEFFICIENT CONJECTURE AND APPLICATIONS

Published online by Cambridge University Press:  01 January 2007

STEPHAN RUSCHEWEYH  and
LUIS SALINAS
STEPHAN RUSCHEWEYH
Affiliation:
Mathematisches Institut, Universität Würzburg, D-97074 Würzburg, Germany e-mail: ruscheweyh@mathematik.uni-wuerzburg.de
LUIS SALINAS
Affiliation:
Departamento de Informática, Universidad Técnica F. Santa María, Valparaíso, Chile e-mail: lsalinas@inf.utfsm.cl
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Abstract.

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D. Brannan's conjecture says that for 0 <α,β≤1, |x|=1, and n∈N one has |A2n−1(α,β,x)|≤|A2n−1(α,β,1)|, where We prove this for the case α=β, and also prove a differentiated version of the Brannan conjecture. This has applications to estimates for Gegenbauer polynomials and also to coefficient estimates for univalent functions in the unit disk that are ‘starlike with respect to a boundary point’. The latter application has previously been conjectured by H. Silverman and E. Silvia. The proofs make use of various properties of the Gauss hypergeometric function.

Keywords

Primary 30C50Secondary 33C0533C45
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 1 , January 2007 , pp. 45 - 52
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

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