In settling the celebrated conjecture of Denjoy on the number of finite asymptotic values of an entire function of finite order, Ahlfors [1) developed a theory which has, subsequently, been applied to problems concerning rates of growth of functions belonging to particular classes. Heins [12] has investigated the smallest members of the class of entire functions having a given number, n, of asymptotic values. In the case n =1, Heins [11] showed that these functions have regular growth, and Kennedy (13) obtained an asymptotic refinement of Ahlfors' Distortion Theorem to prove a corresponding result for n>1. Generalisations of Ahlfors' and Kennedy's results are proved and applied to demonstrate the regularity of growth of the largest functions, regular in the unit disk, which assume a prescribed amount of surface. The growth is, in fact, shown to be radial. Ahlfors' Distortion Theorem was also used by Cartwright [4] to obtain the order of growth of p-valent functions in the unit disk, and by Spencer [15] to extend this to the class of areally mean p-valent functions in the unit disk. By other methods, Hayman [6,7] has shown that the largest circumferentially mean valent functions have regular growth and we prove a similar result for functions mean valent in Spencer's sense. The behaviour of the coefficients of circumferentially mean valent functions has been determined by Hayman [6,7] and, with suitable normalisations, one consequence is a proof of the asymptotic Bieberbach conjecture; we obtain analogous results for the wider mean valent class. Hayman [8] has also proved that the difference between successive moduli of the coefficients of normalised areally mean univalent functions is uniformly bounded and we show that this difference tends asymptotically to zero except in two clearly defined cases.