  Let $\mathcal{L}(\mathscr{H})$ denote the $C^*$-algebra of adjointable
operators on a Hilbert $C^*$-module $\mathscr{H}$. We introduce the generalized
Cauchy-Schwarz inequality for operators in $\mathcal{L}(\mathscr{H})$ and
investigate various properties of operators which satisfy the generalized
Cauchy--Schwarz inequality. In particular, we prove that if an operator
$A\in\mathcal{L}(\mathscr{H})$ satisfies the generalized Cauchy-Schwarz
inequality such that $A$ has the polar decomposition, then $A$ is paranormal.
In addition, we show that if for $A$ the equality holds in the generalized
Cauchy-Schwarz inequality, then $A$ is cohyponormal. Among other things, when
$A$ has the polar decomposition, we prove that $A$ is semi-hyponormal if and
only if $\big\|\langle Ax, y\rangle\big\| \leq
\big\|{|A|}^{1/2}x\big\|\big\|{|A|}^{1/2}y\big\|$ for all $x, y
\in\mathscr{H}$.
