We present a new approach to problems concerning the coefficients (a,) of Bloch and Lipschitz functionsf (z) anZ" (on the unit disk). The approach is based on a new characterization of Bloch functions which is due to Van Casteren [13]. We use an obvious generalization of his result. Some of our results are not new, but the proofs we give are more elegant and straightforward than the existing ones. One of the principal advantages of our approach is that it makes it possible to show the extent to

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... which the results are best-possible. To a large extent, our new results are of this nature. After completing our work, we became aware of unpublished work of A. L. Shields and D. L. Williams which overlaps considerably with ours. We thank them for graciously suggesting that we publish our version. Before stating our principal result we need some terminology. A vector space S of complex sequences (an)n=O (pointwise operations) is called solid if (a.). s S and a;,I _< a, for all n > 0 imply (a,). s S. A function f analytic on the unit disc belongs to the class A (0 < < 1) if (0.1) f'(z)l < c(1/(1 for some constant c > 0. For 0 < < 1, A is the usual Lipschitz class, while A0 is the Bloch space (usually denoted B). 0.2 TH.ORM. The smallest solid (linear) sequence space containing the sequence ofcoefficients (an)n of everyfunctionf(z)= E=o an zn in A (0 < < 1)is the sequence space defined by the condition 2n E a O(n-). j=n In fact, #iven (a.). satisfyin# this condition, there exists a'z A with la, <-la', Yor all n. To get a feeling for this result it may help to consider H (2 < p < ). There, the corresponding solid space is , the square-summable series. This will come as no surprise to those familiar with the standard theorem on random power