Norm Inequalities for Self-Adjoint Derivations

For bounded Hilbert space operators and for all unitarily invariant norms there hold||| |A−B|p|||⩽2p−1|||A |A|p−1−B |B|p−1|||,for all realp⩾2, and ifAandBare additionally self-adjoint, then||| |AX+XB|p|||⩽2p−1 ‖X‖p−1 ||| |A|p−1 AX+XB |B|p−1|||,for all realp⩾3.