Using a refinement of the classical Young inequality, we refine some inequalities of operators including the function $\omega_{p}$, where $% \omega_{p}$ is defined for $p \geqslant 1$ and operators $T_{1}, \ldots, T_{n} \in \mathbb{B}(\mathscr{H})$ by \begin{equation*} \omega_{p}(T_{1}, \ldots, T_{n}):=\sup_{\|x\|=1} \left( \sum_{i=1}^{n} | \left \langle T_{i}x, x \right\rangle|^{p} \right) ^{\frac{1}{p}}. \end{equation*} Among other things, we show that if $T_{1}, \ldots, T_{n} \in \mathbb{B}(% \mathscr{H})$ and $p \geq q \geq 1$ with $\frac{1}{p}+\frac{1}{q} =1,$ then \begin{align*} \frac{1}{n}\left \| \sum_{i=1}^{n}T_{i} \right \|^{2} & \leq \omega_{p}(|T_{1}|, \ldots, |T_{n}|)\omega_{q}(|T_{1}^{*}|, \ldots, |T_{n}^{*}|) \\ & \leq \frac{1}{p} \left\| \sum_{i=1}^{n} | T_{i}|^{p}\right \| + \frac{1}{q} \left\| \sum_{i=1}^{n} | T_{i}^{*}|^{q}\right\| -\inf_{\|x\|=\|y\|=1} \delta(x, y), \end{align*}% where $\delta(x, y) = \frac{1}{p} \left( \sqrt{\sum_{i=1}^{n} \left \langle |T_{i}|x, x \right\rangle^{p} } - \sqrt{\sum_{i=1}^{n} \left \langle |T_{i}^{*}|y, y \right\rangle^{q}} \right)^{2}. $

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