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International Journal of Mathematics and Mathematical Sciences
Volume 17, Issue 1, Pages 193-195

Extensions of Hardy-Littlewood inequalities

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

Received 30 December 1992; Revised 13 May 1993

Copyright © 1994 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.


For a function fHp(Bn), with f(0)=0, we prove

(1) If 0<ps,then01r1(log1r)sβ1Mps(r,Rβf)drfpspfp,s,βp(2) If sp<, thenfp,s,βpfpps01r1(log1r)sβ1Mps(r,Rβf)drwhere Rβf is the fractional derivative of f. These results generalize the known cases s=2, β=1 ([1]).