Criteria for monotonicity of operator means

Source abstract

Let {ψr}r>0 and {ϕr}r>0 be the families of operator monotone functions on [0,∞) satisfying ψr(xrg(x))=xr,ϕr(xrg(x))=xrh(x), where g and h are continuous and g is increasing. Suppose σψa and σ∅r are the corresponding operator connections. We will show that if AσψaB≥1(a>0), then ArσψrB and Arσ∅rB are both increasing for r≥a, and then we will apply this to the geometric operator means to get a simple assertion from which many operator inequalities follow.