Bounds on the derivatives of the Isgur-Wise function with a non-relativistic light quark

In a preceeding study in the heavy quark limit of QCD, it has been demonstrated that the best lower bound on the curvature of the Isgur-Wise function Xsi(w) is Xsi"(1) > (4rho2+3(rho2)2) > 15/16. The quadratic term (rho2)2 is dominant in a non-relativistic expansion in the light quark, both Xsi"(1) and (rho2)2 scaling like (R2m2)2, where m is the light quark mass and R the bound state radius. The non-relativistic limit is thus a good guide-line in the study of the shape of Xsi(w). In the present paper we obtain similar bounds on all the derivatives of XsiNR(w), the IW function with the light quark non-relativistic, and we demons- trate that these bounds are optimal. We formulate a general method, based on the positivity of matrices of moments of the ground state wave function, that allows to bound the n-th derivative XsiNR(n)(w) in terms of the m-th ones (m < n). We argue that this method can be generalized to the true Isgur-Wise function of QCD Xsi(w).