Some properties of the parametric relative operator entropy

The notion of entropy was introduced by Clausius in 1850, and some of the main steps towards the consolidation of the concept were taken by Boltzmann and Gibbs. Since then several extensions and reformulations have been developed in various disciplines with motivations and applications in different subjects, such as statistical mechanics, information theory, and dynamical systems. Fujii and Kamei introduced the notion of the relative operator entropy and Furuta introduced the notion of the parametric relative operator entropy as a generalization of the notion of the relative operator entropy. These concepts can be described by the notion of perspective of some elementary functions and this description makes simplifier their verification. The relative operator entropy is concave and the parametric relative operator entropy is also concave for some suitable parameters. In this paper, we study some properties of the parametric relative operator entropy including its bounds. These bounds generalize the calculated bounds of the relative operator entropy by other authors and in particular the previous results will achieve.