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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 626359, 11 pages
Research Article

Reverses of the Jensen-Type Inequalities for Signed Measures

Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia

Received 26 March 2014; Accepted 8 July 2014; Published 12 August 2014

Academic Editor: Shawn X. Wang

Copyright © 2014 Rozarija Jakšić 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.


In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measure , not necessarily positive, which are generalizations of Jensen's inequality and its reverses for positive measures. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the left-hand and the right-hand side of these inequalities and give several examples of the families of functions for which the obtained results can be applied. The outcome is a new class of Cauchy-type means.