Positive linear maps on normal matrices

For a positive linear map Phi and a normal matrix N, we show that vertical bar Phi(N)vertical bar is bounded by some simple linear combinations in the unitary orbit of Phi(vertical bar N vertical bar). Several elegant sharp inequalities are derived, for instance for the Schur product of two normal matrices A, B is an element of M-n, vertical bar A circle B vertical bar <= vertical bar A vertical bar circle vertical bar B vertical bar + 1/4V (vertical bar A vertical bar circle vertical bar B vertical bar)V* for some unitary V is an element of M-n, where the constant 1/4 is optimal.