Singular value inequalities of matrices via increasing functions

Let A, B, X, and Y be n× n complex matrices such that A is self-adjoint, B≥ 0 , ± A≤ B , max ( X ², Y ² ) ≤ 1 , and let f be a nonnegative increasing convex function on [ 0,∞ ) satisfying $f(0)=0$ . Then 2sⱼ(f ( XAY ) )≤ max \ X ², Y ² \ sⱼ(f(B+A)⊕ f(B-A)) for j=1,2,…,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.