J-selfadjoint matrix means and their indefinite inequalities

Let J be a non trivial involutive Hermitian matrix. Consider \({\mathbb {C}}^n\) equipped with the indefinite inner product induced by J, \([x,y]=y^*J x\) for all \(x,y\in {{\mathbb {C}}}^n,\) which endows the matrix algebra \({\mathbb {C}}^{n\times n}\) with a partial order relation \(\le ^J\) between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint \(\alpha \)-weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type.