Conditions for some polygonal functions to be Bazilevič

Case, B. A.; Quine, J. R.

doi:10.1090/S0002-9939-1983-0695254-0

Conditions for some polygonal functions to be Bazilevič
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by B. A. Case and J. R. Quine PDF

Proc. Amer. Math. Soc. 88 (1983), 257-261 Request permission

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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