‎In this paper‎, ‎we prove that if Phi‎: mathscr{A} to mathscr{B} is a bounded linear map between C^*-algebras‎, ‎then‎ begin{equation*}‎ left|sum_{k=1}^n left{|Phi(A_k)|^2+|Phi(A_k^*)|^2right}right|leq 2 |Phi|^2 left | sum_{k=1}^n‎ ‎ left( |A_k|^2+|A_k^*|^2 right) right |‎ ‎ end{equation*}‎ ‎for all $n in mathbb{N}$ and all $A_1, ldots‎, ‎A_n in mathscr{A}$‎. ‎This improves the Haagerup--Pisier--Ringrose (H-P-R) inequality for Hermitian bounded maps Phi‎. ‎In addition‎, ‎we present a refinement of the H-P-R inequality and give an example to support it‎. ‎Moreover‎, ‎we establish several versions of the H-P-R inequality involving unitarily invariant norms‎. ‎Further‎, ‎we present some useful inequalities in the setting of Hilbert C^*-modules and then apply it to get several H-P-R type inequalities‎.

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