Zeros of A ^p functions and related classes of analytic functions

The function f(z), analytic in the unit disc, is in A ^p if {Mathematical expression}. A necessary condition on the moduli of the zeros of A ^p functions is shown to be best possible. The function f(z) belongs to B ^p if {Mathematical expression}. Let {z _n } be the zero set of a B ^p function. A necessary condition on |z _n | is obtained, which, in particular, implies that Σ(1-|z _n |)^1+(1/p)+g <∞ for all ε>0 (p≧1). A condition on the Taylor coefficients of f is obtained, which is sufficient for inclusion of f in B ^p. This in turn shows that the necessary condition on |z _n | is essentially the best possible. Another consequence is that, for q≧1, p<q, there exists a B ^p zero set which is not a B ^q zero set.