Coefficients and integral means of some classes of analytic functions

Sheil-Small, T.

doi:10.1090/S0002-9939-1983-0695258-8

Coefficients and integral means of some classes of analytic functions
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by T. Sheil-Small PDF

Proc. Amer. Math. Soc. 88 (1983), 275-282 Request permission

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 ${(1 + xz)^\alpha }{(1 - z)^{ - \beta }} \ll {(1 + z)^\alpha }{(1 - z)^{ - \beta }}$ $(\left | x \right | = 1,\alpha \geqslant 1,\beta \geqslant 1)$ 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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