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Coefficients and integral means of some classes of analytic functions


Author: T. Sheil-Small
Journal: Proc. Amer. Math. Soc. 88 (1983), 275-282
MSC: Primary 30C45; Secondary 30C50
DOI: https://doi.org/10.1090/S0002-9939-1983-0695258-8
MathSciNet review: 695258
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Abstract: The sharp coefficient bounds for the classes $ {V_k}$ of functions of bounded boundary rotation are obtained by a short and elementary argument. Elementary methods are also applied for the coefficients of related classes characterised by a generalised Kaplan condition. The result $ {(1 + xz)^\alpha }{(1 - z)^{ - \beta }} \ll {(1 + z)^\alpha }{(1 - z)^{ - \beta }}$ $ (\left\vert x \right\vert = 1,\alpha \geqslant 1,\beta \geqslant 1)$ is proved simply. It is further shown that the functions $ {(1 + z)^\alpha }{(1 - z)^{ - \beta }}$ are extremal for the $ p$th means ($ p$ an arbitrary real) of all Kaplan classes $ K(\alpha ,\beta )$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0695258-8
Keywords: Functions of bounded boundary rotation, starlike functions, close-to-convex functions
Article copyright: © Copyright 1983 American Mathematical Society

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