Distances between critical points and midpoints of zeros of hyperbolic polynomials

Author(s)

Dimitrov, Dimitar K. ^[1] ; Kostov, Vladimir P. ^[2]

Affiliation

^[1] Univ Estadual Paulista, IBILCE, Dept Ciencias Computacao & Estatist, BR-15054000 Sao Jose Do Rio Preto, SP - Brazil

^[2] Univ Nice, Math Lab, F-06108 Nice 2 - France

• Document type: Journal article
• Source: BULLETIN DES SCIENCES MATHEMATIQUES; v. 134, n. 2, p. 196-206, MAR 2010.
• Web of Science Citations: 1

Abstract

Let p(x) be a polynomial of degree n with only real zeros x(1) <= x(2) <= ... <= x(n). Consider their midpoints z(k) = (x(k) + x(k+1))/2 and the zeros xi(1) <= xi(2) <= ... <= xi(n-1) of p'(z). Motivated by a question posed by D. Farmer and R. Rhoades, we compare the smallest and largest distances between consecutive xi(k) to the ones between consecutive z(k). The corresponding problem for zeros and critical points of entire functions of order one from the Laguerre-Polya class is also discussed. (C) 2007 Published by Elsevier Masson SAS. (AU)

• FAPESP's process: 03/01874-2 - Orthogonal and similar polynomials: properties and applications
• Grantee: Alagacone Sri Ranga
• Support Opportunities: Research Projects - Thematic Grants