Zeros of functions in the Bergman spaces

A function/(z) analytic in the unit disk is said to belong to the Bergman space A p (0ffî\f(re ie )\ p rdr d6 =H co , the space of bounded analytic functions in the disc.Assuming that /(O) 5*0, we list the zeros of/in order of nondecreasing modulus: 0<|z 1 |^|z 2 |^|z 3 |^-• -0,LEMMA 2. IfS { ^ = 0(N a )for some a^l, then ƒ G A P for allp<2\tL.LEMMA 3.For some p, l^p^2, suppose that 2jv-i#~*»S'Ar ) < 00 an ^ N*-*S% } =0(1).ThenfeA p \ l//?+l// = l.Lemma 1 is proved by an application of Jensen's theorem.Lemmas 2 and 3 follow from corresponding coefficient conditions, after a summation by parts.In particular, Lemma 3 is a consequence of the fact that AMS (MOS) subject classifications (