Singular value inequalities of matrices via increasing functions

Let A, B, X, and Y be $n\times n$ $n\times n$ complex matrices such that A is self-adjoint, $B\ge 0$ $B\geq 0$ , $\pm A\le B$ $\pm A\leq B$ , $max({\parallel X\parallel }^2,{\parallel Y\parallel }^2)\le 1$ $\max ( \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} ) \leq 1$ , and let f be a nonnegative increasing convex function on $[0,\mathrm{\infty })$ $[ 0,\infty ) $ satisfying $f(0)=0$ $f(0)=0$ . Then $2s_j\left(f\left(\right|XA{Y}^{\ast }\left|\right)\right)\le max\{{\parallel X\parallel }^2,{\parallel Y\parallel }^2\}{s}_j(f(B+A)\oplus f(B-A))$ \( 2s_{j}\bigl(f \bigl( \bigl\vert XAY^{\ast } \bigr\vert \bigr) \bigr)\leq \max \bigl\{ \Vert X \Vert ^{2}, \Vert Y \Vert ^{2} \bigr\} s_{j}\bigl(f(B+A)\oplus f(B-A)\bigr) \) for $j=1,2,\dots ,n$ $j=1,2,\ldots,n$ . This singular value inequality extends an inequality of Audeh and Kittaneh. Several generalizations for singular value and norm inequalities of matrices are also given.