矩阵的柯西-施瓦茨不等式的进一步改进
Further refinements of the Cauchy-Schwarz inequality for matrices
摘要：
设$A, B$和$X$是$n\times n$矩阵，使得$A, B$是半正定的。利用积分技巧和Hermite-Hadamard不等式的各种改进，给出了矩阵柯西-施瓦茨不等式的一些改进.特别地，我们建立了不等式\begin{align*} |||\,|A^{1\over2}XB^{1\over2}|^r|||^2&\leq|||\,|A^{t}XB^{1-s}|^r||| \,\,\,|||\,|A^{1-t}XB^{s}|^r|||\\& \leq\max \{|||\,|AX|^r||| \,\,\,|||\,|XB|^r|||,|||\,|AXB|^r||| \,\,\,|||\,|X|^r|||\}, \end{align*}，其中$s,t\in[0,1]$和$r\geq0$ .
英文摘要：
Let $A, B$ and $X$ be $n\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard inequality. In particular, we establish the inequality \begin{align*} |||\,|A^{1\over2}XB^{1\over2}|^r|||^2&\leq|||\,|A^{t}XB^{1-s}|^r||| \,\,\,|||\,|A^{1-t}XB^{s}|^r|||\\& \leq\max \{|||\,|AX|^r||| \,\,\,|||\,|XB|^r|||,|||\,|AXB|^r||| \,\,\,|||\,|X|^r|||\}, \end{align*} where $s,t\in[0,1]$ and $r\geq0$.
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