Advanced refinements of Young and Heinz inequalities

Abstract

In this article, we prove several multi-term refinements of Young type inequalities for both real numbers and operators improving several known results. Among other results, we prove that for all 0≤ν≤1 and each N∈N,A#νB+∑j=1Nsj(ν)(A#αj(ν)B+A#21−j+αj(ν)B−2A#2−j+αj(ν)B)≤A∇νB, for the positive operators A and B, where sj(ν) and αj(ν) are certain functions. Moreover, some new Heinz type inequalities involving the Hilbert–Schmidt norm are established.