Some singular value inequalities via convexity

If $c_1(Z)\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge c_n(Z)$ denote the Euclidean lengths of the column vectors of any $n\times n$ matrix $Z,$ then a fundamental inequality related to Hadamard products states that $$\sum _{i=1}^k{\sigma }_i(X^{\ast }Y\circ B)\le \sum _{i=1}^k{c}_i(X)c_i(Y){\sigma }_i(B)1\le k\le n,$$ where ${\sigma }_i(\cdot )$ is the $i$th singular value. In this paper, we shall offer a simple proof of this result via convexity arguments. In addition, this technique is applied to obtain some further singular value inequalities as well.