The lower bound on independence number

Let G be a graph with degree sequence (d(v)). If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least Sigma(v)f(m+1)(d(v)), where f(m+1) (x) is a function greater than log(x/(m+1))-1/x for x>0. For a weighted graph G = (V, E, w), we prove that its weighted independence number (the maximum sum of the weights of an independent set in G) is at least Sigma(v)w(v)/1+d(v), where w, is the weight of v.