Multidimensional Jensen’s operator on a Hilbert space and applications

Motivated by a joint concavity of connections, solidarities and multidimensional weighted geometric mean, in this paper we extend an idea of convexity (concavity) to operator functions of several variables. With the help of established definitions, we introduce the so called multidimensional Jensen’s operator and study its properties. In such a way we get the lower and upper bounds for the above mentioned operator, expressed in terms of non-weighted operator of the same type. As an application, we obtain both refinements and converses for operator variants of some well-known classical inequalities. In order to obtain the refinement of Jensen’s integral inequality, we also consider an integral analogue of Jensen’s operator for functions of one variable.