Notes on majorizations for singular values

Abstract

. In this note, we mainly investigate the majorizations on the products and sums of matrices. Firstly, we present the following result: Let A i , B i and X i ∈ M n ( C ) ( i = 1 , 2 , ··· , m ) with X i ( i = 1 , 2 , ··· , m ) are invertible matrices, and let h be a nonnegative increasing continuous function on [ 0 , + ∞ ) with h ( 0 ) = 0. If f , g are nonnegative continuous functions on [ 0 , + ∞ ) with f ( t ) g ( t ) = t for t ∈ [ 0 , + ∞ ) , then