Some extensions of the Young and Heinz inequalities for Matrices

Abstract: In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two positive semidefinite matrices $A$ and $B$ we show that \begin{align*} \Big|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\Big|{2}^{2}\leq\Big|AX+XB\Big|{2}^{2}- 2r\Big|AX-XB\Big|{2}^{2}-r{0}\left(\Big|A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big|_{2}^{2}+ \Big|A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big|_{2}^{2}\right), \end{align*} where $X$ is an arbitrary $n\times n$ matrix, $0<\nu\leq\frac{1}{2}$, $r=\min{\nu, 1-\nu}$ and $r_{0}=\min{2r, 1-2r}$.