A note on indefinite chaotic order

A selfadjoint involutive matrix J endows C_n with an indefinite inner product [.,.]given by [x; y] := y^*Jx, x,y in C_n. For a pair of J-selfadjoint matrices A, B with positive eigenvalues,LogA > =J LogB is called the J-chaotic order. In this note a characterization of the indefinite chaoticorder in terms of the alpha-power mean is revisited and a related open problem is proposed. A new characterization of the indefinite chaotic order is also obtained.