Completely positive mappings and mean matrices

Some functions f:R+→R+ induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define a linear mapping on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of in the case of a few concrete functions f. This problem has been motivated by applications in quantum information.