Journal of Inequalities and Applications
Volume 7 (2002), Issue 5, Pages 727-746
doi:10.1155/S1025583402000371
Weighted integral inequalities with the geometric mean operator
1Department of Mathematics, Luleå University, Luleå S-97 187, Sweden
2Computer Centre, Russian Academy of Sciences, Far-Eastern Branch, Khabarovsk 680042, Russia
Received 20 March 2001; Revised 29 June 2001
Copyright © 2002 Lars-Erik Persson and Vladimir D. Stepanov. 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
The geometric mean operator is defined by
Gf(x)=exp(1x∫0xlogf(t)dt).
A precise two-sided estimate of the norm
‖G‖=supf≠0‖G‖Luq‖f‖Lvp
for 0<p, q≤∞ is given and some applications and extensions are pointed out.