Bounds on the derivatives of the Isgur-Wise function from sum rules in the heavy quark limit of QCD

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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 jᵖ 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, we obtain sum rules involving the derivatives of the elastic Isgur-Wise function ξ(w) at zero recoil, that imply that the n-th derivative can be bounded by the (n-1)-th one. For the curvature σ² = ξ''(1), this proves the already proposed bound σ² ≥ 5 4 ρ². Moreover, we obtain the absolute bound for the n-th derivative (-1)ⁿ ξ⁽ⁿ⁾(1) ≥ (2n+1)!! 2²ⁿ, that generalizes the results ρ² ≥ 3 4 and σ² ≥ 15 16.