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International Journal of Mathematics and Mathematical Sciences
Volume 16, Issue 1, Pages 205-207
http://dx.doi.org/10.1155/S0161171293000250

Generalizations of inequalities of littlewood and paley

Mathematics Department, Qufu Normal University, Shandong, Qufu 273165, China

Received 30 July 1991; Revised 13 December 1991

Copyright © 1993 Hindawi Publishing Corporation. 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.

Abstract

For a function f, holomorphic in the open unit ball Bn in Cn, with f(0)=0, we prove (I) If 0<s2 and sp< Then fppC01Bn|f(ρζ)|ps|Rf(ρζ)|s(log1/ρ)s1ρ1dσ(ζ)dρ (ii) If 2Bp< Then 01Bn|f(ρζ)|ps|Rf(ρζ)|s(log1/ρ)s1ρ1dσ(ζ)dρCfpp where Rf is the radial dervative of f, generalizing the known cases p=s([1]) and p=s, n=1 ([2]).