‎The Haagerup--Pisier--Ringrose inequality states that if $\\Phi$ is a bounded linear map from a $C^*$-algebra $\\mathscr{A}$ into a $C^*$-algebra $\\mathscr{B}$‎, ‎then‎

‎\\begin{align}\\label{HPR}‎

‎\\left\\|\\sum_{j=1}^n...
|\\cdot...
|$‎, ‎$\\Phi‎: ‎\\mathscr{A} \\to \\mathscr{B}$ is a bounded linear map between $C^*$-algebras‎, ‎$A_1‎, ‎\\cdots‎, ‎A_n\\in \\mathscr{A}$ are positive such that $\\Phi(A_j)$'s commute for $1\\leq j\\leq n$ and $X_1‎, ‎\\cdots‎, ‎X_n \\in \\mathcal{J}$‎. ‎Then‎

‎\\begin{align*}‎

‎\\left|\\left|\\left| \\sum_{j=1}^n\\left\\{ \\Phi(A_...
|X_1 \\oplus \\cdots\\oplus X_n...
|\\‎, ‎\\|\\Phi\\|^2\\left\\|\\sum_{j=1}^n A_j^2\\right\\|‎.

‎\\end{align*}‎...

‎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 ...