  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 $r\geq0$.
