Singular value inequalities of matrices via increasing functions

Abstract

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