Zero sets of holomorphic functions in the unit ball with slow growth

We study the zero-varieties of holomorphic functions in the unit ball satisfying the growth condition log |f(z)|≤c fλ(|z|), where λ:(0,1)→ℝ+ is a positive increasing function. We obtain some sufficient conditions on an analytic variety to be defined by such a function. Some results for the particular case λ(r)=log(e/(1-r)), corresponding to the class A -∞, generalize those of B. Korenblum in one variable. © 1995 The Magnes Press, The Hebrew University.