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

Lars-Erik Persson1 and Vladimir D. Stepanov2

 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

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.