Some mappings related to Levinson’s operator inequality and its converses

The goal of my talk is to consider the construction of mappings related to Levinson's inequality and its converses for Hilbert space operators. To obtain these results, appropriate mappings related to Jensen's operator inequality are given. As application, refinements of some know inequalities are obtained. \ The starting point for my research are references: \ [1] S. S. Dragomir, Inequalities for Functions of Selfadjoint Operators on Hilbert Spaces, Victoria University, Melbourne, 2011. \ [2] J. Micic, J. Pecaric and M. Praljak, Levinson's inequality for Hilbert space operators, J. Math. Inequal. 9 (2015), 1271-1285. \ [3] J. Micic and J. Pecaric, On Levinson's operator inequality and its converses, J. Inequal. Appl. 2015 (2015), 1-15. \ [4] J. Jaksetic, J. Pecaric and M. Praljak, A monotonic refinement of Levinson’s inequality, J. Inequal. Appl., 2015 (2015) 162, 1-7.