‎Let $T\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathfrak{B}(\\\\\\\\\\\\\\\\mathbb{H})$ be a bounded linear operator and $\\\\\\\\\\\\\\\\omega(T)$ denotes the numerical radius of $T$‎. ‎Let $S_1,\\\\\\\\\\\\\\\\ldots‎, ‎S_m,T_1,\\\\\\\\\\\\\\\\ldots,T_m\\\\\\\\\\\\\\\\in\\\\\\\\\\\\\\\\mathfrak{B}(\\\\\\\\\\\\\\\\mathbb{H})$ such that $S_iT_j=T_jS_i$ and $S_i^*T_j=T_jS_i^*$ for all $1\\\\\\\\\\\\\\\\leq i,j,\\\\\\\\\\\\\\\\leq m$‎. ‎It is shown that‎ ‎\\\\\\\\\\\\\\\\begin{equation*}‎ ‎\\\\\\\\\\\\\\\\omega\\\\\\\\\\\\\\\\left(\\\\\\\\\\\\\\\\sum_{i=1}^{m}S_iT_i\\\\\\\\\\\\\\\\right) \\\\\\\\\\\\\\\\leq\\\\\\\\\\\\\\\\frac{1}{2} \\\\\\\\\\\\\\\\left\\\\\\\\\\\\\\\\|\\\\\\\\\\\\\\\\sum_{i=1}^{m} S_i^*S_i+S_iS_i^*\\\\\\\\\\\\\\\\right\\\\\\\\\\\\\\\\|^{\\\\\\\\\\\\\\\\frac{1}{2}}\\\\\\\\\\\\\\\\left\\\\\\\\\\\\\\\\|\\\\\\\\\\\\\\\\sum_{i=1}^{m} T_i^*T_i+T_iT_i^*\\\\\\\\\\\\\\\\right\\\\\\\\\\\\\\\\|^{\\\\\\\\\\\\\\\\frac{1}{2}}‎. ‎\\\\\\\\\\\\\\\\end{equation*}‎ ‎Moreover‎, ‎we show that‎ ‎\\\\\\\\\\\\\\\\begin{equation*}‎ ‎\\\\\\\\\\\\\\\\omega\\\\\\\\\\\\\\\\left(\\\\\\\\\\\\\\\\sum_{i=1}^{m}S_iT_i+T_iS_i\\\\\\\\\\\\\\\\right) \\\\\\\\\\\\\\\\leq \\\\\\\\\\\\\\\\left\\\\\\\\\\\\\\\\|\\\\\\\\\\\\\\\\sum_{i=1}^{m} S_i^*S_i+S_iS_i^*\\\\\\\\\\\\\\\\right\\\\\\\\\\\\\\\\|^{\\\\\\\\\\\\\\\\frac{1}{2}}\\\\\\\\\\\\\\\\left\\\\\\\\\\\\\\\\|\\\\\\\\\\\\\\\\sum_{i=1}^{m} T_i^*T_i+T_iT_i^*\\\\\\\\\\\\\\\\right\\\\\\\\\\\\\\\\|^{\\\\\\\\\\\\\\\\frac{1}{2}}‎, ‎\\\\\\\\\\\\\\\\end{equation*}‎ ‎for all $S_1,\\\\\\\\\\\\\\\\ldots‎, ‎S_m,T_1,\\\\\\\\\\\\\\\\ldots,T_m\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathfrak{B}(\\\\\\\\\\\\\\\\mathbb{H})$‎. ‎In particular‎, ‎for $m=1$‎, ‎we refine some well known inequalities

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