tailieunhanh - Báo cáo hóa học: " Research Article Schur-Type Inequalities for Complex Polynomials with no Zeros in the Unit Disk"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Schur-Type Inequalities for Complex Polynomials with no Zeros in the Unit Disk | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 90526 10 pages doi 2007 90526 Research Article Schur-Type Inequalities for Complex Polynomials with no Zeros in the Unit Disk Szilard Gy. Revesz Received 20 March 2007 Accepted 28 June 2007 Recommended by Saburou Saitoh Starting out from a question posed by T. Erdelyi 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. Erdelyi 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 Erdelyi 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. Copyright 2007 Szilard Gy. Revesz. 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 .