In this paper we consider a certain order-like relation for self-adjoint operators on a Hilbert space. This relation is defined by using the Jensen inequality. We will show that under some assumptions this relation is antisymmetric.
"A note on the Jensen inequality for self-adjoint operators." J. Math. Soc. Japan 62 (3) 949 - 961, July, 2010. https://doi.org/10.2969/jmsj/06230949