This is a continuation of our previous paper. We consider a certain order-like relation for positive operators on a Hilbert space. This relation is defined by using the Jensen inequality with respect to the square-root function. We show that this relation is antisymmetric if the operators are invertible.
Tomohiro HAYASHI. "A note on the Jensen inequality for self-adjoint operators, II." J. Math. Soc. Japan 66 (1) 111 - 121, January, 2014. https://doi.org/10.2969/jmsj/06610111