© 1989 by London Mathematical Society
On Extremal Functions for John Constants
Department of Mathematical Sciences, Michigan Technological University Houghton, Michigan 49931, USA
We define the John constant 
(D) of a domain D 
C to be sup{
1:1 
|f'(z)| in D implies that f 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 f0 is called an extremal function for such a D if 1 |f'0(z)| (D) on D and f0(a)=fo(b) for two distinct points a,b
D. A simple compactness argument shows that there exist extremal functions for H. Let BN(D, ) = {f:f' = eh, where Re{h} is the harmonic measure of N the union of N arcs on 
D}. It is shown that f0 is an extremal function for D, then f BN(D, In (D)) for some N. As a corollary we deduce that for any such D there exists K such that (D) = sup{ex: all f in BK (D, ) are univalent in D}; in particular this holds when D is the unit disk.