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Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

Published online by Cambridge University Press:  20 November 2018

W. C. Royster
W. C. Royster*
Affiliation:
University of Kentucky
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Let Σ represent the class of analytic functions

(1)

which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion

(2).

In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 540 - 551
Copyright
Copyright © Canadian Mathematical Society 1962

References

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