Refinements of operator Cauchy–Schwarz and Minkowski inequalities for p-modified norms and related norm inequalities

For (degree) p  -modifications Φ(p)Φ(p) of symmetrically norming functions Φ and finite families of bounded Hilbert space operators A1,B1,⋯,AN,BNA1,B1,⋯,AN,BN and X‖∑n=1NAn⁎XBn‖Φ(p)≤‖(|∑n=1NAn⁎XBn|2+∑n=1N|An(‖∑n=1NAn⁎An‖12I+(‖∑n=1NAn⁎An‖I−∑n=1NAn⁎An)12)−1∑m=1NAm⁎XBm−‖∑n=1NAn⁎An‖12XBn|2)p2‖Φ1p=‖∑n=1NAn⁎An‖12‖(∑n=1NBn⁎X⁎XBn)12‖Φ(p) for 0