Using the OPE and the trace formalism, we have obtained a number of sum rules in the heavy quark limit of QCD that include the sum over all excited states for any value j P of the light cloud. We show that these sum rules imply that the elastic Isgur-Wise function ξ(w) is an alternate series in powers of (w − 1). Moreover, one gets that any n-th derivative of ξ(w) at w = 1 can be bounded by the (n − 1)-th one. For the curvature σ 2 = ξ ′′ (1), this proves the already proposed bound σ 2 ≥ 5 4 ρ 2 . Moreover, we obtain the absolute lower bound for the n-th derivative (−1) n ξ (n) (1) ≥ (2n+1)!! 2 2n , that generalizes the results ρ 2 ≥ 3 4 and σ 2 ≥ 15 16 .