Some inequalities of Glaeser-Bronštein type

The classical Glaeser estimate is a special case of the Lemma of Bronshtein which states the Lipschitz continuity of the roots of a hyperbolic polynomial whose coefficients depend on a real parameter. Here we prove a pointwise estimate for the successive derivatives of the coefficients of the polynomial in term of certain nonnegative functions which are symmetric polynomials of the roots (hence also of the coefficients). These inequalities are very helpful in the study of the Cauchy problem for homogeneous weakly hyperbolic equations of higher order.