Advanced refinements of Young and Heinz inequalities

• A#νB+∑j=1Nsj(ν)(A#αj(ν)B+A#21−j+αj(ν)B−2A#2−j+αj(ν)B)≤A∇νB.
• Following the same theme of this inequality, we prove reversed versions.
• Norm inequalities involving the Heinz means are studied.

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≤10≤ν≤1 and each N∈NN∈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)sj(ν) and αj(ν)αj(ν) are certain functions. Moreover, some new Heinz type inequalities involving the Hilbert–Schmidt norm are established.