linear preserves of logarithm majorization

Let x, y ∈ r n , x, y > 0, we say x logarithm majorized by y , written x ≺log y if log x ≺ log y . let m+ nm be the collection of matrices with positive entries. for x, y ∈ m+ nm, it is said that x is logarithm column (row) majorized by y , and is denoted as x ≺ column log y (x ≺ row log y ), if xj ≺log yj (xi ≺log yi) for all j = 1, 2, · · · m(i = 1, 2, · · · n), where xj and yj (xi and yi) are the ith column (row) of x and y respectively. in the present paper, the relations column (row) logarithm majorization on m+ nm are studied and also all linear operators t : m+ nm −→ m+ nm preserving column (row) logarithm majorization will be characterized.