# Some inequalities of Glaeser-Bronšteĭn type

### Giovanni Taglialatela

Università degli Studi di Bari, Italy### Sergio Spagnolo

Università di Pisa, Italy

## 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.