International Journal of Mathematics and Mathematical Sciences 
Volume 31 (2002), Issue 2, Pages 85-95
doi:10.1155/S016117120201311X

Opial type Lp-inequalities for fractional derivatives

G. A. Anastassiou,1 J. J. Koliha,2 and J. Pečarić3

 1Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
2Department of Mathematics and Statistics, University of Melbourne, Melbourne, VIC 3010, Australia
3Faculty of Textile Technology, University of Zagreb, Zagreb 10 000, Croatia
Received 7 May 2001; Revised 18 December 2001

Abstract

This paper presents a class of Lp-type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author for continuous functions (1998). The novelty of our approach is the use of the index law for fractional derivatives in lieu of Taylor's formula, which enables us to relax restrictions on the orders of fractional derivatives.