An operator monotone function image and strictly chaotic order Original Research Article

We consider the function f(t) defined by image Firstly we shall show direct and simplified proofs of operator monotonicity of f(t) and its dual function image without appealing of Löwner Theorem. Then we shall show the following: image By using the result (*), we summarize the following interesting contrast between A>B>0 and Agreater-or-equal, slantedB>0. image image Agreater-or-equal, slantedB>0 implies f(Aα)greater-or-equal, slantedf(Bα) forall αset membership, variant ]0,1].Finally we cite a counterexample related to (*) as follows: image