On Extremal Functions For John Constants

Document Type

Article

Publication Date

1-1-1989

Abstract

We define the John constant y(D) of a domain D cz C to be sup (a > 1:1 < /'(z)l ^a> nD implies that/is univalent in D), and consider the case in which D is the upper half-plane H or a Jordan domain with sufficiently smooth boundary. A function /0is called an extremal function for such a D if 1 ^ /o(z)l ^ y(D) on D and f0(a)=f0(b) for two distinct points a, bedD. A simple compactness argument shows that there exist extremal functions for H. Let BN(D, a) = (/:/' = e(XA, where Re(/i) is the harmonic measure of the union of N arcs on dD). It is shown that if/0is an extremal function for D, then/e BN(D, In y(D)) for some N. As a corollary we deduce that for any such D there exists K such that y(D) = sup(e“: All /in BK(D, a) are univalent in D); in particular this holds when D is the unit disk. © 1989, Oxford University Press. All rights reserved.

Publication Title

Journal of the London Mathematical Society

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