@article { author = {Ulukök, Z.}, title = {Singular values of convex functions of matrices}, journal = {Bulletin of the Iranian Mathematical Society}, volume = {43}, number = {6}, pages = {2057-2066}, year = {2017}, publisher = {Iranian Mathematical Society (IMS)}, issn = {1017-060X}, eissn = {1735-8515}, doi = {}, abstract = {‎Let $A_{i},B_{i},X_{i},i=1,\dots,m,$ be $n$-by-$n$ matrices such that $‎\sum_{i=1}^{m}\left\vert A_{i}\right\vert ^{2}$ and $‎\sum_{i=1}^{m}\left\vert B_{i}\right\vert ^{2}$  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 $f\left( 0\right)‎ ‎=0 $‎, ‎then  $$‎2s_{j}\left( f\left( \frac{\left\vert \sum_{i=1}^{m}A_{i}^{\ast‎ ‎ }X_{i}B_{i}\right\vert }{\sqrt{\left\Vert \sum_{i=1}^{m}\left\vert‎ ‎ A_{i}\right\vert ^{2}\right\Vert \left\Vert \sum_{i=1}^{m}\left\vert‎ ‎ B_{i}\right\vert ^{2}\right\Vert }}\right) \right) \leq s_{j}\left( \oplus‎ ‎_{i=1}^{m}f\left( 2X_{i}\right) \right)‎$$ ‎for $j=1,\ldots,n$‎. ‎Applications of our results are given.}, keywords = {Singular value,arithmetic-geometric mean,direct sum,positive semidefinite matrix,convex function}, url = {http://bims.iranjournals.ir/article_1083.html}, eprint = {http://bims.iranjournals.ir/article_1083_327a7a884377fbd11d819f33653250c0.pdf} }