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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

An asymptotic formula for an integral in starlike function theory


Authors: R. R. London and D. K. Thomas
Journal: Trans. Amer. Math. Soc. 215 (1976), 393-406
MSC: Primary 30A32
MathSciNet review: 0387563
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Abstract: The paper is concerned with the integral

$\displaystyle H = \int _0^{2\pi }\vert f{\vert^\sigma }\vert F{\vert^\tau }{(\operatorname{Re} F)^\kappa }\;d\theta $

in which f is a function regular and starlike in the unit disc, $ F = zf'/f$, and the parameters $ \sigma ,\tau ,\kappa $ are real. A study of H is of interest since various well-known integrals in the theory, such as the length of $ f(\vert z\vert = r)$, the area of $ f(\vert z\vert \leqslant r)$, and the integral means of f, are essentially obtained from it by suitably choosing the parameters. An asymptotic formula, valid as $ r \to 1$, is obtained for H when f is a starlike function of positive order $ \alpha $, and the parameters satisfy $ \alpha \sigma + \tau + \kappa > 1,\tau + \kappa \geqslant 0,\kappa \geqslant 0,\sigma > 0$. Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.

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DOI: https://doi.org/10.1090/S0002-9947-1976-0387563-0
Keywords: Starlike function, order, length, area, integral means
Article copyright: © Copyright 1976 American Mathematical Society