Various results are known concerning the rate of growth of the derivative of a function f(z), analytic and univalent in the circle Izi <1; among the simplest of these results we mention Koebe's Verzerrungsatz (cf. [4, p. 86]), according to which If'(z) =0[(1IzI)-3] in IzI <1. However, as rapid a rate of growth as is permitted by this theorem can be attained only on a relatively small set of radii; indeed, Seidel and Walsh [5] have shown that for all eti on I z = 1, except for a possible set of linear measure zero,