Let M be a semi-finite von Neumann algebra with a faithful normal semifinite trace \(\tau \) , and let \(L_{0}(M)\) be the set of all \(\tau \) -measurable operators. On this space, \(\mu _{t}(A)\) \((t>0)\) denotes the generalized singular numbers of \(A\in L_{0}(M)\) . In this work, we prove the following inequality: if \(f:[0,\infty )\rightarrow [0,\infty )\) is a continuous concave function, for any \(A_{i}\in L_{0}(M)\) \((1\le i\le m)\) , \(\begin{aligned} \mu \left( f\left( \left| \sum _{i=1}^{m}A_{i}\right| \right) \right) \preccurlyeq \mu \left( f\left( \frac{1}{2}\left( \begin{array}{cc} \sum _{i=1}^{m}\left| A_{i}\right| & \sum _{i=1}^{m}A_{i}^{*} \\ \sum _{i=1}^{m}A_{i} & \sum _{i=1}^{m}\left| A_{i}^{*}\right| \end{array} \right) \right) \right) \preccurlyeq \sum _{i=1}^{m}\mu \left( f\left( \left| A_{i}\right| \right) \right) . \end{aligned}\)
