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

DOI:
https://doi.org/10.1090/S0002-9939-1983-0695254-0

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 's is specified such that the function is Bazilevič for some . For any interval where , a function of this type which is Bazilevič precisely when is in this interval is found. Examples are given of non-Bazilevič functions with polygonal images and Bazilevič functions which are Bazilevič for a single value .

**[1]**A. Baernstein II,*Integral means, univalent functions and circular symmetrization*, Acta Math.**133**(1974), 139-169. MR**0417406 (54:5456)****[2]**I. E. Bazilevič,*On a case of integrability in quadratures of the Löwner-Kufarev equation*, Mat. Sb.**37**(1955), 471-476. (Russian) MR**0072949 (17:356e)****[3]**D. Campbell and K. Pearce,*Generalized Bazilevič functions*, Rocky Mountain J. Math.**9**(1979), 197-226. MR**519937 (80d:30005)****[4]**J. B. Conway,*Functions of one complex variable*, Springer-Verlag, New York, 1978. MR**503901 (80c:30003)****[5]**Z. Nehari,*Introduction to complex analysis*, rev. ed., Allyn and Bacon, Boston, Mass., 1968. MR**0224780 (37:379)****[6]**J. C. Plaster,*A simple geometric criterion for non-Bazilevičness*, Proc. Amer. Math. Soc. (to appear).**[7]**T. Sheil-Small,*On Bazilevič functions*, Quart, J. Math. Oxford Ser. (2)**23**(1972), 135-42. MR**0299799 (45:8847)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0695254-0

Keywords:
Univalent function,
Bazilevič function

Article copyright:
© Copyright 1983
American Mathematical Society