Embedding of analytic function spaces with given mean growth of the derivative

MSC (2000) 30D55 If φ is a positive function defined in (0,1) and 0 <p< ∞, we consider the space L(p, φ) which consists of all functions f analytic in the unit disc D for which the integral means of the derivative Mp(r, f � )= " 1 2π Rπ −π þ f � (re iθ ) þ p dθ "1/p , 0 <r< 1, satisfy M p(r, f � )=O (φ(r)) ,a sr → 1. In this paper, for any given p ∈ (0,1), we characterize the functions φ, among a certain class of weight functions, to be able to embedd L(p, φ) into classical function spaces. These results complement other previously obtained by the authors for p ≥ 1. c