A note on monotonic and semi-monotonic matrix functions

A real valued function f on is called monotonic if A+  B+ implies f(A) ≤ f(B). Here A+ is the matrix obtained by taking the absolute values of the entries of A. And A+ ≤ B+ is to be understood entrywise If A+ ≤ B+ implies f(A) ≤ f(B) then f is called semi-monoionic. In this note we show that some of the best known matrix functions are semi-monotonic, but not monotonic.