  The baryon differential spectrum of the baryon decay $\Lambda_b \to \Lambda_c
\ell \bar{\nu}_\ell$ will be measured in detail at LHCb. We obtain new results
on the form factors in the heavy quark expansion of Heavy Quark Effective
Theory that can be useful in the interpretation of the data. We formulate a sum
rule for the elastic subleading form factor $A(w)$ at order $1/m_Q$, that
originates from the Lagrangian perturbation $\mathcal{L}_{kin}$. In the sum
rule appear only the intermediate states $(j^P, J^P) = (0^+, {1 \over 2}^+)$,
entering also in the $1/{m_Q^2}$ correction to the axial form factor $G_1(w)$,
that contributes to the differential rate at zero recoil $w = 1$. This result,
together with another sum rule in the forward direction for $|G_1(1)|^2$,
allows us to obtain a lower bound for the correction at zero recoil $-
\delta_{1/{m_Q^2}}^{(G_1)}$ in terms of the derivative $A'_1(1)$ and the slope
$\rho^2_\Lambda$ and curvature $\sigma^2_\Lambda$ of the elastic Isgur-Wise
function $\xi_{\Lambda}(w)$. Another theoretical implication is that $A'(1)$
must vanish for some relation between $\rho^2_\Lambda$ and $\sigma^2_\Lambda$,
as well as for $\rho^2_\Lambda \to 0$, establishing a non-trivial correlation
between the leading IW function $\xi_{\Lambda}(w)$ and the subleading one
$A(w)$. A phenomenological estimation of these two functions allows to obtain a
lower bound on $- \delta_{1/{m_Q^2}}^{(G_1)}$.
