The Turán number of sparse spanning graphs

For a graph H, the extremal numberex(n,H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H. Let @d(H)>0 and @D(H) denote the minimum degree and maximum degree of H, respectively. We prove that for all n sufficiently large, if H is any graph of order n with @D(H)=