Relationships between partial orders of matrices and their powers

Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices <f>A</f> and <f>B</f> is related to the corresponding order between the squares <f>A2</f> and <f>B2</f>, in the sense of the star partial ordering, the minus partial ordering, and the Lo¨wner partial ordering. In the present paper, possibilities of generalizing and strengthening their results are studied from two points of view: by widening the class of matrices considered and by replacing the squares by arbitrary powers. [Copyright &y& Elsevier]