Some recent results about Jensen’s operator inequality and its converses with application to operator means

The goal of this lecture is to present a generalization of Jensen's operator inequality and its converses for fields of positive linear mappings such that is not unital. First we give general formulations of Jensen's operator inequality and it's converses. As a consequence, difference and ratio type of converses of Jensen's operator inequality are obtained. In addition, we discuss the order among quasi-arithmetic means in a general setting. As an application we get some comparison theorems for power functions and power means. Also we give the chaotic order version of these results.