Table of Contents
International Scholarly Research Notices
Volume 2014, Article ID 580605, 12 pages
http://dx.doi.org/10.1155/2014/580605
Research Article

Improvements of Jensen-Type Inequalities for Diamond- Integrals

1School of Natural Sciences, National University of Sciences and Technology, Islamabad 44000, Pakistan
2Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia

Received 29 March 2014; Accepted 7 May 2014; Published 14 August 2014

Academic Editor: Jan L. Cieśliński

Copyright © 2014 Rabia Bibi 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 give further improvements of the Jensen inequality and its converse on time scales, allowing also negative weights. These results generalize the Jensen inequality and its converse for both discrete and continuous cases. Further, we investigate the exponential and logarithmic convexity of the differences between the left-hand side and the right-hand side of these inequalities and present several families of functions for which these results can be applied.