Let
${{A}_{i}},\,{{B}_{i}}$
and
${{X}_{i}}\,(i\,=\,1,\,2,\ldots ,\,n)$
be operators on a separable Hilbert space. It is shown that if
$f$
and
$g$
are nonnegative continuous functions on
$\left[ 0,\infty \right)$
which satisfy the relation
$f\,(t)g(t)\,=\,t$
for all
$t$
in
$\left[ 0,\infty \right)$
, then
$${{\left\| \left| \,{{\left| \sum\limits_{i=1}^{n}{A_{i}^{*}{{X}_{i}}{{B}_{i}}} \right|}^{r}} \right| \right\|}^{2}}\,\le \,\left\| \left| {{\left( \sum\limits_{i=1}^{n}{A_{i}^{*}f{{(\left| X_{i}^{*} \right|)}^{2}}{{A}_{i}}} \right)}^{r}} \right| \right\|\,\,\left\| \left| {{\left( \sum\limits_{i=1}^{n}{B_{i}^{*}g{{(\left| {{X}_{i}} \right|)}^{2}}\,{{B}_{i}}} \right)}^{r}} \right| \right\|$$
for every
$r\,>\,0$
and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.