J-selfadjoint matrix means and their indefinite inequalities

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