Journal of Inequalities and Applications 
Volume 4 (1999), Issue 4, Pages 327-338
doi:10.1155/S1025583499000430

Landau and Kolmogoroff type polynomial inequalities

Claudia R. R. Alves and Dimitar K. Dimitrov

 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

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.