A (g, f)-factor F of a graph G is Called a Hamiltonian (g, f)-factor if F contains a Hamiltonian cycle. The binding number of G is defined by $bind(G);=;{min};{;{frac{{mid}N_GX{mid}}{{mid}X{mid}}};{mid};{emptyset};{ eq};X;{subset};V(G)},;{N_G(X);{ eq};V(G)};}$. Let G be a connected graph, and let a and b be integers such that $4;{leq};a;<;b$. Let g, f be positive integer-valued functions defined on V(G) such that $a;{leq};g(x);<;f(x);{leq};b$ for every $x;{in};V(G)$. In this paper, it is proved that if $bind(G);{geq};{frac{(a+b-5)(n-1)}{(a-2)n-3(a+b-5)},};{ u}(G);{geq};{frac{(a+b-5)^2}{a-2}}$ and for any nonempty independent subset X of V(G), ${mid};N_{G}(X);{mid};{geq};{frac{(b-3)n+(2a+2b-9){mid}X{mid}}{a+b-5}}$, then G has a Hamiltonian (g, f)-factor.