The author studies the optimal order of convergence of Hermite-Fejér interpolation for general systems of nodes. The first result in this direction is given by the author himself in 1973. Later, Xin-Long Zhou extended this result to the case when
is the matrix of Jacobi nodes with non-positive parameters. Also, Shi worked on this order in 1991. This was the first step in proving that all Hermite-Fejér interpolations are saturated with at most of order
. In this paper the author makes another step by extending and strengthening the result of Shi to all polynomials.