Weighted integrals of higher order derivatives of analytic functions

In this paper we work with the class of differentiable weights w in the unit disc D such that (sup)o<r<i w'(r)/w(r)(2) integral(1)(r) w(x) dx <infinity and w'(r)/w(r)(2) integral(1)(r) w(x)dx >= -1 0<r<1 We prove that if w is one of these weights, N is a positive integer, and 0 < p < infinity, 0 < q <= infinity, then the equivalence integral M-1(0)q(P)(r, f)w(r) dr asymptotic to sup(vertical bar z vertical bar<1/2) vertical bar f (z)vertical bar(p) + integral M-1(0)q(p) (r,f((N)))(psi(w)(r))(NP) w(r) dr, holds for all analytic functions f in D. The above result generalizes a classical equivalence due to Flett and extends a previous result of the authors to derivatives of higher order. We also extend a result of the first author and prove some results on Hadamard products.