Abstract
Starting from the results given in [J. Pecaric, I. Peric, M. Rodic Lipanovic: Uniform treatment of Jensen type inequalities, to appear in Math. Rep. (Bucur.)] where the uniform treatment of the Jensen type inequalities and its converses is given, we investigate the exponential convexity of differences of the left-hand and the right-hand side of these inequalities. Using these differences, we produce new exponentially convex functions. Finally, we give several examples of the families of functions for which the obtained results can be applied, and we get some generalized Cauchy means. Results from this paper present the generalization of the results from [M. Anwar, J. Pecaric: Cauchy's means for signed measures, Bull. Malays. Math. Sci. Soc. (2) 34(1) (2011), 31-38.].
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