Journal of Inequalities and Applications
Volume 2005 (2005), Issue 5, Pages 583-591

Weighted integrals of holomorphic functions in the unit polydisc

Stevo Stević

Mathematical Institute of Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia

Received 28 September 2003

Copyright © 2005 Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let f be a measurable function defined on the unit polydisc Un in Cn and let ωj(zj), j=1,,n, be admissible weights on the unit disk U, with distortion functions ψj(zj), ω,Np,q(Un)={f|fω,Np,q<}, where fω,Np,qq=[0,1)nMpq(f,r)Πj=1nωj(rj)drj, and 𝒜ω,Np,q(Un)=ω,Np,q(Un)H(Un). We prove the following result: if p,q[1,) and for all j=1,,n, ψj(zj)(f/zj)(z)ω,Np,q, then f𝒜ω,Np,q and there is a positive constant C=C(p,q,ωj,n) such that f𝒜ω,Np,qC(|f(0)|+j=1nψj(f/zj)ω,Np,q).