On a double parameterized Relative Operator Entropy and related properties

Since its genesis, the concept of entropy has not ceased to invade enough domains. Its different uses have led to various definitions of this notion and to several generalizations. In this article, we work in the same direction. Precisely, the main objective of this paper is to provide a generalization to the relative operator entropy defined in 1989 by J. I. Fujii and E. Kamei for positive operators. We focus on establishing some algebraic properties and on determining certain inequalities involving the new generalized relative operator entropy. By investing some techniques from functional calculus, we were able to establish results generalizing certain properties of the relative operator entropy known in the literature.