For a connected undirected graph G = (V;E) with vertex set {1; 2;: :: ; n} and degrees di, for 1≤ i≤ n, we show that ABC(G)≤ → (n-1)(|E|-R-1(G)); where R-1(G) = ∑ (i;j)εE 1/didj is the Randić index. This bound allows us to obtain some maximal results for the ABC index with elementary proofs and to improve all the upper bounds in [20], as well as some in [17], using lower bounds for R-1(G) found in the literature and some new ones found through the application of majorization.