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ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the coefficients of functions with bounded boundary rotation


Author: D. K. Thomas
Journal: Proc. Amer. Math. Soc. 36 (1972), 123-129
MSC: Primary 30A34
DOI: https://doi.org/10.1090/S0002-9939-1972-0308384-2
MathSciNet review: 0308384
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Abstract: Let $ {V_k}$ be the class of normalised functions of bounded boundary rotation. For $ f \in {V_k}$ define

$\displaystyle M(r,f) = \mathop {\max }\limits_{\vert z\vert = r} \vert f(z)\vert,$

and let $ L(r,f)$ denote the length of $ f(\vert z\vert = r)$. Then if $ f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}} $, it is shown that (i) $ 2M(r,f) < L(r,f) \leqq k\pi M(r,f)$, and (ii) $ {n^2}\vert{a_n}\vert \leqq (3k/{r^{n - 1}})M(r,f'),n \geqq 2$. The class $ {\Lambda _k}$ of meromorphic functions of boundary rotation is also studied and estimates for the coefficients are given.

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DOI: https://doi.org/10.1090/S0002-9939-1972-0308384-2
Article copyright: © Copyright 1972 American Mathematical Society

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