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Proceedings of the American Mathematical Society

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Conditions for some polygonal functions to be Bazilevič

Authors: B. A. Case and J. R. Quine
Journal: Proc. Amer. Math. Soc. 88 (1983), 257-261
MSC: Primary 30C45
MathSciNet review: 695254
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Abstract: Univalent functions in the disc whose image is a particular eight-sided polygonal region determined by two parameters are studied. Whether such a function is Bazilevič is determined in terms of the two parameters, and the set of real $ \alpha $'s is specified such that the function is $ (\alpha ,\beta )$ Bazilevič for some $ \beta $. For any interval $ \left[ {a,b} \right]$ where $ 1 < a \leqslant 3 \leqslant b$, a function of this type which is $ (\alpha ,0)$ Bazilevič precisely when $ \alpha $ is in this interval is found. Examples are given of non-Bazilevič functions with polygonal images and Bazilevič functions which are $ (\alpha ,0)$ Bazilevič for a single value $ \alpha $.

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Keywords: Univalent function, Bazilevič function
Article copyright: © Copyright 1983 American Mathematical Society

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