Some inequalities of Glaeser-Bronšteĭn type

Abstract

The classical Glaeser estimate is a special case of the Lem\-ma of Bron{\v{s}}te{\u{\i}}n which states the Lipschitz continuity of the roots \( \la_j(x) \) of a hyperbolic polynomial $P(x,X)$ with coefficients $a_j(x)$ depending on a real parameter. Here we prove a pointwise estimate for the successive derivatives of the $a_j(x)$'s in term of certain nonnegative functions which are symmetric polynomials of the roots \( \{\la_j(x)\} \) (hence also of the coefficients $\{a_j(x)\})$. These inequalities are very helpful in the study of the Cauchy problem for homogeneous weakly hyperbolic equations of higher order.