Levinson's type generalization of the Jensen inequality and its converse for real Stieltjes measure

Rozarija Mikić; Josip Pečarić; Mirna Rodić

doi:10.1186/s13660-016-1274-y

Levinson's type generalization of the Jensen inequality and its converse for real Stieltjes measure

J Inequal Appl. 2017;2017(1):4. doi: 10.1186/s13660-016-1274-y. Epub 2017 Jan 3.

Authors

Rozarija Mikić¹, Josip Pečarić¹, Mirna Rodić¹

Affiliation

¹ Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, Zagreb, 10000 Croatia.

Abstract

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi's inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results.

Keywords: Giaccardi’s inequality; Green function; Hermite-Hadamard’s inequality; Jensen’s inequality; Levinson’s inequality; converse Jensen’s inequality; mean-value theorems.