Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 948430, 17 pages
doi:10.1155/2010/948430
Research Article

Generalization of an Inequality for Integral Transforms with Kernel and Related Results

Sajid Iqbal,1 J. Pečarić,1,2 and Yong Zhou3

1Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan
2Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
3School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China

Received 27 March 2010; Revised 2 August 2010; Accepted 27 October 2010

Academic Editor: András Rontó

Copyright © 2010 Sajid Iqbal et al. 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

We establish a generalization of the inequality introduced by Mitrinović and Pečarić in 1988. We prove mean value theorems of Cauchy type for that new inequality by taking its difference. Furthermore, we prove the positive semidefiniteness of the matrices generated by the difference of the inequality which implies the exponential convexity and logarithmic convexity. Finally, we define new means of Cauchy type and prove the monotonicity of these means.