Abstract. In the present paper, an approach for constructing new means in two variables is investigated. Application of this approach for proving some mean-inequalities is also discussed.Mathematics subject classification (2010): 26E60.

Abstract. Let A and B be two accretive operators. We first introduce the weighted geometric mean of A and B together with some related properties. Afterwards, we define the relative entropy as well as the Tsallis entropy of A and B. The present definitions and their related results extend those already introduced in the literature for positive invertible operators.

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We investigate some results about mean-inequalities involving a large number of bivariate means. As application, we derive a lot of inequalities between four or more means among the standard means known in the literature.