Criteria for monotonicity of operator means

Abstract

Let {ψ_r}_r>0 and {φ_r}_r>0 be the families of operator monotone functions on [0, ∞) satisfying ψ_r(x^rg(x))/x^r, φ_r(x^rg(x))/x^rh(x), where g and h are continuous and g is increasing. Suppose σ_{ψ_a} and σ_{φ_r} are the corresponding operator connections. We will show that if A^aσ_{ψ_a}B≥q 1 (a>0), then A^rσ_{ψ_r}B and Aσ_{φ_r}B are both increasing for r≥q a, and then we will apply this to the geometric operator means to get a simple assertion from which many operator inequalities follow.