Abstract. In this paper we work with the class of differentiable weights $\omega $ in the unit disc ${\msbm D}$ such that $$\sup_{0< r<1}{\omega '(r)\over\omega (r)^2}\int_r^1\omega(x) dx < \infty $$ and $${\omega '(r)\over\omega (r)^2}\int_r^1\omega(x) dx \ge -1, 0< r< 1. $$ We prove that if $\omega $ is one of these weights, $N$ is a positive integer, and $0< p< \infty,$ $0< q\le\infty $, then the equivalence $$ \int_0^1 M_q^p(r,f)\omega(r) dr \asymp\sup _{|z|< 1/2}|f(z)|^p+ \int_0^1 M_q^p(r,f^{(N)})(\psi_\omega(r))^{Np} \omega(r) dr, $$ holds for all analytic functions $f$ in ${\msbm D}$. 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.

AMS Subject Classification (1991): 30D55, 32A36, 46E15

Keyword(s): Weighted integrals, sucessive derivatives, Hadamard products

Received August 11, 2005. (Registered under 5909/2009.)