Mean growth of the derivative of analytic functions, bounded mean oscillation, and normal functions

For a given positive function phi defined in [0, 1) and 1 less than or equa l to p < infinity, we consider the space L(p, phi) which consists of all fu nctions f analytic in the unit disc Delta for which (1/2 pi integral(-pi)(pi)\f' (re(i theta))\(p) d theta)(1/p) = O (phi(r)), as r --> 1. A result of Bourdon, Shapiro and Sledd implies that such a space is contain ed in BMOA for phi(r) = (1 - r)(1/p-1). Among other results, in this paper we prove that this result is sharp in a very strong sense, showing that, fo r a large class of weight functions phi, the function phi(r) = (1 - r)(1/p- 1) is the best one to get L(p, phi) subset of BMOA. Actually, if phi(r)(1 - r)(1-1/p) up arrow infinity, as r up arrow 1, we construct a function f is an element of L(p, phi) which is not a normal function. These results impr ove other obtained recently by the second author. We also characterize the functions phi, among a certain class of weight functions, to be able to emb edd L(p, phi) into H-q for q > p or into the space B of Bloch functions.