An extension of the weighted geometric mean in unital JB-algebras

Let \({\mathcal {A}}\) be a unital JB-algebra and \(A,B\in {\mathcal {A}}\). The weighted geometric mean \(A\sharp _r B\) for \(A,B\in {\mathcal {A}}\) has been recently defined for \(r\in [0,1].\) In this work, we extend the weighted geometric mean \(A\sharp _r B\), from \(r\in [0,1]\) to \(r\in (-1, 0)\cup (1, 2)\). We will notice that many results will be reversed when the domain of r change from [0, 1] to \((-1,0)\) or (1, 2). We also introduce the Heinz and Heron means of elements in \({\mathcal {A}}\), and extend some known inequalities involving them.