  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 $\sigma^2 \geq {15 \over 16}$.
