Singular values of convex functions of matrices

‎Let $A_{i},B_{i},X_{i},i=1,dots,m,$ be $n$-by-$n$ matrices such that $‎sum_{i=1}^{m}leftvert A_{i}rightvert ^{2}$ and $‎sum_{i=1}^{m}leftvert B_{i}rightvert ^{2}$  are nonzero matrices and each $X_{i}$ is‎ ‎positive semidefinite‎. ‎It is shown that if $f$ is a nonnegative increasing ‎convex function on $left[ 0,infty right) $ satisfying $fleft( 0right)‎ ‎=0 $‎, ‎then  $$‎2s_{j}left( fleft( frac{leftvert sum_{i=1}^{m}A_{i}^{ast‎ ‎ }X_{i}B_{i}rightvert }{sqrt{leftVert sum_{i=1}^{m}leftvert‎ ‎ A_{i}rightvert ^{2}rightVert leftVert sum_{i=1}^{m}leftvert‎ ‎ B_{i}rightvert ^{2}rightVert }}right) right) leq s_{j}left( oplus‎ ‎_{i=1}^{m}fleft( 2X_{i}right) right)‎$$ ‎for $j=1,ldots,n$‎. ‎Applications of our results are given.