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