Szilárd Gy. Révész

Copyright © 2007 Szilárd Gy. Révész. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Starting out from a question posed by T. Erdélyi and J. Szabados, we consider Schur-type inequalities for the classes of complex algebraic polynomials having no zeros within the unit disk D. The class of polynomials with no zeros in D—also known as Bernstein or Lorentz class—was studied in detail earlier. For real polynomials utilizing the Bernstein-Lorentz representation as convex combinations of fundamental polynomials (1−x)k(1+x)n−k, G. Lorentz, T. Erdélyi, and J. Szabados proved a number of improved versions of Schur- (and also Bernstein- and Markov-) type inequalities. Here we investigate the similar questions for complex polynomials. For complex polynomials, the above convex representation is not available. Even worse, the set of complex polynomials, having no zeros in the unit disk, does not form a convex set. Therefore, a possible proof must go along different lines. In fact, such a direct argument was asked for by Erdélyi and Szabados already for the real case. The sharp forms of the Bernstein- and Markov-type inequalities are known, and the right factors are worse for complex coefficients than for real ones. However, here it turns out that Schur-type inequalities hold unchanged even for complex polynomials and for all monotonic, continuous weight functions. As a consequence, it becomes possible to deduce the corresponding Markov inequality from the known Bernstein inequality and the new Schur-type inequality with logarithmic weight.