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

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 $\xi (w)$ is an alternate series in powers of $(w-1)$. Moreover, we obtain sum rules involving the derivatives of the elastic Isgur-Wise function $\xi (w)$ at zero recoil, that imply that the $n$-th derivative can be bounded by the $(n-1)$-th one. For the curvature $\sigma^2 = \xi''(1)$, this proves the already proposed bound $\sigma^2 \geq {5 \over 4} \rho^2$. Moreover, we obtain the absolute bound for the $n$-th derivative $(-1)^n \xi^{(n)}(1) \geq {(2n+1)!! \over 2^{2n}}$, that generalizes the results $\rho^2 \geq {3 \over 4}$ and $\sigma^2 \geq {15 \over 16}$.