Some recent results about Levinson’s operator inequality and its converses

The goal of my talk is to present Levinson's inequality and its converses for Hilbert space operators. At first we give general formulations of Levison's operator inequality and its converses for some large class of "3-convex functions at a point c" and normalized positive linear mappings. Consequently, order among quasi-arithmetic operator means is obtained. In addition, we consider the construction of mappings related to Levinson's inequality. Some properties of these mappings are observed: operator convexity, monotonicity and quasi linearity. To obtain these results, we give appropriate mappings related to the Davis-Choi- Jensen operator inequality. Moreover, we consider mappings related to converses of Levinson's operator inequality. As applications, refinements and extensions of some know inequalities are obtained.