On the zeros of Bloch functions

A function f, analytic in the unit disc Delta, is said to be a Bloch functi on if sup (z is an element of Delta)(1 - \Z\(2))\f'(z)\ < infinity. In this paper we study the zero sequences of non-trivial Bloch functions. among ot her results we prove that if f is a Bloch function with f(0) + 0 and {z(k)} is the sequence of ordered zeros of f, then Pi(k=1)(N) 1/(\z(k)\) = O((log N)(1/2)), as N --> infinity (i) and Sigma(\zk\ > 1 - 1/e) (1 - \z(k)\)(log log 1/(1 - \z(k)\))(-alpha) < infini ty, for all alpha > 1. (ii) We will also prove that (ii) is best possible even for the little Bloch spa ce B-0. To this end we construct a function f is an element of B-0 whose ze ro sequence {z(k)} satisfies Sigma(\zk\ > 1 - 1/e) (1 - \z(k)\)(log log 1/(1 - \z(k)\))(-1) = infinity. We also consider analogous problems for some other related spaces of analyt ic functions.