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

C*-module operators which satisfy in the generalized Cauchy--Schwarz type inequality