Advanced refinements of Young and Heinz inequalities

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 \begin{eqnarray*} A\#_νB&+&\sum_{j=1}^{N}s_{j}(ν)\left(A\#_{α_j(ν)}B+A\#_{2^{1-j}+α_j(ν)}B-2A\#_{2^{-j}+α_j(ν)}B\right)\leq A\nabla_νB, \end{eqnarray*} for the positive operators $A$ and $B$, where $0\leq ν\leq 1, N\in\mathbb{N}$ and $α_j(ν)$ is a certain function. Moreover, some new Heinz type inequalities involving the Hilbert-Schmidt norm are established.

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