Let G be a 2 connected graph of order n, and let a and b be integers such that 2≤a＜b, and let g(x) and f(x) be two nonnegative integer valued functions defined on V(G) such that a≤g(x)＜f(x)≤b for each x∈V(G). It is proved that G has a Hamiltonian (g,f) factor if the minimum degree of G satisfies the following conditions,δ(G)≥[(b-1)^{2}-(a-1)(b-a)]/(a-1)〖SX)〗,[n＞(a+b-3)(a+b-2)]/(a-1), and max｛d_{G(x) },d_{G(y)} ｝≥(b-1)n/(a+b-2) for any two nonadjacent vertices x and y in G.