Journal of Inequalities and Applications
Volume 4 (1999), Issue 4, Pages 327-338
doi:10.1155/S1025583499000430
Landau and Kolmogoroff type polynomial inequalities
Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, São José do Rio Preto 15054-000, SP, Brazil
Received 4 December 1998; Revised 2 February 1999
Copyright © 1999 Claudia R. R. Alves and Dimitar K. Dimitrov. 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
Let 0<j<m≤n be integers. Denote by ‖⋅‖ the norm ‖f‖2=∫−∞∞f2(x)exp(−x2)dx. For various positive values of A and B we establish Kolmogoroff type inequalities
‖f(j)‖2≤A‖f(m)‖+B‖f‖Aθk+Bμk,
with certain constants θkeμk, which hold for every f∈πn (πn denotes the space of real algebraic polynomials of degree not exceeding n).
For the particular case j=1 and m=2, we provide a complete characterisation of the positive constants A and B, for which the corresponding Landau type polynomial inequalities
‖f′‖2≤A‖f″‖+B‖f‖Aθk+Bμk,
hold. In each case we determine the corresponding extremal polynomials for which equal-
ities are attained.