JournalsrlmVol. 17 , No. 4DOI 10.4171/rlm/474

Some inequalities of Glaeser-Bronšteĭn type

  • Giovanni Taglialatela

    Università degli Studi di Bari, Italy
  • Sergio Spagnolo

    Università di Pisa, Italy
Some inequalities of Glaeser-Bronšteĭn type cover

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 \laj(x)\la_j(x) of a hyperbolic polynomial P(x,X)P(x,X) with coefficients aj(x)a_j(x) depending on a real parameter. Here we prove a pointwise estimate for the successive derivatives of the aj(x)a_j(x)'s in term of certain nonnegative functions which are symmetric polynomials of the roots {\laj(x)}\{\la_j(x)\} (hence also of the coefficients {aj(x)})\{a_j(x)\}). These inequalities are very helpful in the study of the Cauchy problem for homogeneous weakly hyperbolic equations of higher order.