Inequality between unitary orbits

For bounded self-adjoint operators A and B we write A <= B if there is a unitary U such that A <= U* BU. In [7], Kosaki (1992) has shown that A <=, B =t exp(A) < exp(B). In this note, we extend this; especially, we show that for a function f(t) = Sigma(n)(i=1)C(i)t(a1). e(bit) with positive coefficients a(i), b(i) and c(i), 0 <= A <= B f (A) <= f (B). We then apply this to a positive linear map and get a similar inequality. (C) 2013 Published by Elsevier Masson SAS on behalf of Academie des sciences.