Adjointations of Operator Inequalities and Characterizations of Operator Monotonicity via Operator Means

We propose adjointations between operator orderings, which convert any operator inequalities/identities associated with certain binary operations to new ones. Then we prove that a continuous function f : (0, infinity) -> (0, infinity) is operator monotone increasing if and only if f (A !(t) B) <= f(A) !(t) f(B) for any positive operators A, B and scalar t is an element of [0,1]. Here, !(t) denotes the t-weighted harmonic mean. As a counterpart, f is operator monotone decreasing if and only if the reverse of preceding inequality holds. Moreover, we obtain many characterizations of operator monotone increasingness/decreasingness in terms of operator means. These characterizations lead to many operator inequalities involving means