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The underlying problem confronting us in this paper is the determination of precise upper bounds for the moduli of the coefficients in these normalized expansions at oo for functions in the classes S and T. This problem for the first two coefficients in each class has already been solved. We note first that since 4) and f are inverse, a1 = b1 and a2 =-b2, so for these two cases, discussion of only one of the two classes is necessary. For higher coefficients, the a's and b's are related by certain polynomials, so in general, a separate study must be made for each class. Gronwall [6](1), Bieberbach [1], and Faber [3] showed that I a, ?1 with the unique function

The coefficient problem for schlicht mappings of the exterior of the unit circle