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

In a preceding 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 xi(w) is xi"(1)>1/5[4rho(2)+3(rho(2))(2)]>15/16. The quadratic term (rho(2))(2) is dominant in a nonrelativistic expansion in the light quark, both xi"(1) and (rho(2))(2) scaling like (R(2)m(q)(2))(2), where m(q) is the light quark mass and R the bound state radius. The nonrelativistic limit is thus a good guideline in the study of the shape of xi(w). In the present paper we obtain similar bounds on all the derivatives of xi(NR)(w), the IW function with the light quark nonrelativistic, and we demonstrate that these bounds are optimal. Our general method is based on the positivity of matrices of moments of the ground state wave function, that allows to bound the nth derivative xi(NR)((n))(w) in terms of the mth ones (m<n). We show that the method can be generalized to the true Isgur-Wise function of QCD xi(w).