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Transactions of the American Mathematical Society

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



Asymptotic behavior of functions with bounded boundary rotation

Author: James W. Noonan
Journal: Trans. Amer. Math. Soc. 164 (1972), 397-410
MSC: Primary 30A32
MathSciNet review: 0294618
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Abstract: For $ k \geqq 2$ denote by $ {V_k}$ the class of normalized functions, analytic in the unit disc, which have boundary rotation at most $ k\pi $. Let $ {a_n}$ be the nth Taylor coefficient of $ f(z) \in {V_k}$. Let $ {I_\lambda }(r,f')$ and $ {I_\lambda }(r,f)$ be the $ \lambda $-integral mean of $ f'(z)$ and $ f(z)$ respectively. We determine asymptotic formulas for $ f'(z)$, and these formulas are then applied to study the behavior of $ \vert{a_n}\vert$ as $ n \to \infty $, and the behavior of $ {I_\lambda }(r,f')$ and $ {I_\lambda }(r,f)$ as $ r \to 1$.

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Keywords: Asymptotic behavior, bounded boundary rotation, convex functions, starlike functions, coefficients, integral mean
Article copyright: © Copyright 1972 American Mathematical Society

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