JournalsrlmVol. 17, No. 4pp. 367–375

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
Download PDF

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.

Cite this article

Giovanni Taglialatela, Sergio Spagnolo, Some inequalities of Glaeser-Bronšteĭn type. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 4, pp. 367–375

DOI 10.4171/RLM/474