Monotonicity of order preserving operator functions Original Research Article

We discuss monotonicity of order preserving operator functions and related order preserving operator inequalities. Let Agreater-or-equal, slantedBgreater-or-equal, slanted0 with A>0,tset membership, variant[0,1] and pgreater-or-equal, slanted1. Letimage We show that: (i) F(r,w)greater-or-equal, slantedF(r,1)greater-or-equal, slantedF(r,s)greater-or-equal, slantedF(r,s′)for any s′greater-or-equal, slantedsgreater-or-equal, slanted1,rgreater-or-equal, slantedt and image, (ii) F(q,s)greater-or-equal, slantedF(t,s)greater-or-equal, slantedF(r,s)greater-or-equal, slantedF(r′,s)for any r′greater-or-equal, slantedrgreater-or-equal, slantedt,sgreater-or-equal, slanted1 andt-1less-than-or-equals, slantqless-than-or-equals, slantt. These imply the following recent inequality due to Kameiimage