Growth estimates for conformal mappings and for positive harmonic functions in space

Abstract

This thesis falls naturally into two distinct parts. Both come under the general heading of the theory of conformal mapping though the later part incorporates work in potential theory. We first study the growth of means of the logarithmic derivative of a univalent function in the disc. Here results have been obtained by Hayman and by Baernstein and Brown. Hayman has shown that an elementary upper bound for the growth of these means is best possible in general. Later on, Baernstein and Brown showed that the means of certain classes of monotone slit mappings, including support points of the class S of normalised univalent functions, grow no faster than those of the Koebe function up to a multiplicative constant. The question remained open for unrestricted monotone slit mappings. We settle this question by constructing a monotone slit mapping the mean of whose logarithmic derivative grows faster than that of the Koebe function. Following this, we discuss some recent work by Burdzy on the boundary behaviour of positive harmonic functions in Lipschitz domains and applications of this work to the angular derivative problem. Burdzy obtains his results on the angular derivative by probabilistic methods. Rodin and Warschawski later gave a classical proof of part of Burdzy’s main result and related his criteria for the existence of an angular derivative to criteria which they had used previously. They were, however, unable to obtain a non-probabilistic proof of the full theorem. Using a new non-probabilistic method, we prove a theorem on the growth of positive harmonic functions vanishing near a boundary point of a Lipschitz domain. The plane case of this result and some special cases in space were proved by Burdzy in a series of articles. He went on to prove the full result in space in a later paper with R. J. Williams. Our result enables us to give an elementary proof of the remainder of Burdzy’s theorem on the angular derivative and so complements Rodin and Warschawski’s work. We complete our study of this problem by proving two further related results on the boundary behaviour of positive harmonic functions in Lipschitz domains. It is likely that the methods used will be helpful in problems of a similar nature.