An equivalence for weighted integrals of an analytic function and ist derivative

In this paper we introduce the class of differentiable weights ω in the unit disc 𝔻 such that equation image where L is a positive constant. The main result in this paper asserts that if ω is one of these weights, then the equivalence equation image holds for all 0 < p < ∞, 0 < q ≤ ∞and f an analytic function in 𝔻. Our results improve others due to Aleman, Siskakis, and Stević. We also prove two results on harmonic conjugate functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)