The John constant \(\gamma\) (D) of a domain D is the infimum of \(\alpha\) such that every analytic f satisfying \(1\leq | f'(z)| \leq \alpha\) for all \(z\in D\) is univalent in D. If \(1\leq | f'(z)| \leq \gamma (D)\) and \(f(a)=f(b)\) for some \(a,b\in \partial D\), f is extremal for D. The author shows that, under broad assumptions, the derivative of an extremal is of the form \(e^{\alpha h}\), where Re h is the harmonic measure of the union of a finite number of arcs on \(\partial D\). Consequently, the extremals can be sought from the collection of functions whose derivatives are of the kind described.