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On functions of bounded boundary rotation


Author: Ming-chit Liu
Journal: Proc. Amer. Math. Soc. 29 (1971), 345-348
MSC: Primary 30.42
DOI: https://doi.org/10.1090/S0002-9939-1971-0286993-6
MathSciNet review: 0286993
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Abstract: Let $ U = \{ z = r{e^{i\theta }}\left\vert {r < 1\} } \right.$. For $ k \geqq 2$ let $ {V_k}$ be the class of normalized analytic functions $ f(z)$ such that the boundary rotation of $ f(U)$ is at most $ k\pi $. Let $ A(r)$ be the integral

$\displaystyle \int_0^{2\pi } {\int_0^r {\left\vert {f'(\rho {e^{i\theta }})} \right\vert} } {^2}\rho d\rho d\theta ,$

$ L(r)$ the length of the image of the circle $ \left\vert z \right\vert = r$ under the mapping $ f(z)$. In this paper the author proves that for $ z \in U$ if $ f(z) \in {V_k}$ then

$\displaystyle \mathop {\lim \sup }\limits_{r \to 1} \left( {\mathop {{\operator... ...i A(r)\log \left( {\frac{{1 + r}}{{1 - r}}} \right)} \right)^{ - 1/2}} \leqq k.$

This generalizes to arbitrary $ k \geqq 2$ the recent result of Nunokawa for the case $ k = 2$.

References [Enhancements On Off] (What's this?)

  • [1] K. Löwner, Untersuchungen über die Verzerrung bei konformen Abbildungen des Einheitskreises $ \left\vert z \right\vert < 1$, die durch Funktionen mit nicht verschwindender Ableitung geliefert werden, Ber. Kön. Sächs. Ges. Wiss. Leipzig 69 (1917), 89-106.
  • [2] M. Nunokawa, A note on convex and Bazilevič functions, Proc. Amer. Math. Soc. 24 (1970), 332-335. MR 40 #4437. MR 0251206 (40:4437)
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  • [5] -, Univalent functions $ f(z)$ for which $ zf'(z)$ is spirallike, Michigan Math. J. 16 (1969), 97-101. MR 39 #5785. MR 0244471 (39:5785)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0286993-6
Keywords: Analytic mapping, function of bounded boundary rotation, convex function, curve length, order of infinity
Article copyright: © Copyright 1971 American Mathematical Society

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