Cauchy-Schwarz norm inequalities for weak∗-integrals of operator valued functions

For a σ-finite measures μ on Ω and μ-weakly∗-measurable families {At}t∈Ω and {Bt}t∈Ω of Hilbert space operators we have the non-commutative Cauchy-Schwarz inequalities in Schatten p-ideals ∫ΩAXBdμp⩽∫ΩA∗∫ΩAA∗dμq−1Adμ2qX∫ΩB∫ΩB∗Bdμr−1B∗dμ2rpfor all X∈p(H) and for all p,q,r⩾1 such that 1q+1r=2p. If both {At}t∈Ω and {Bt}t∈Ω consists of commuting normal operators, then∫ΩAXBdμ⩽∫ΩA∗AdμX∫ΩB∗Bdμfor all unitarily invariant norms |||·||| and all X∈|||·|||(H). If additionally ∫ΩA∗Adμ⩽I and ∫ΩB∗Bdμ⩽I, then I−∫ΩA∗AdμXI−∫ΩB∗Bdμ∈|||·|||(H) andI−∫ΩA∗AdμXI−∫ΩB∗Bdμ⩽X−∫ΩAXBdμ.Applications include Young's and arithmetic-geometric-logarithmic means inequalities for operators and the mean value theorem for operator monotone functions.