Landau and Kolmogoroff Type Polynomial Inequalities

Let 0 < j < m ≤ n be integers. Denote by ‖ · ‖ the norm ‖f‖2 = ∫∞ −∞ f 2(x) exp(−x2)dx. For various positive values of A and B we establish Kolmogoroff type inequalities ‖f ‖ ≤ A‖f (m)‖2 +B‖f‖2 Aθk +Bμk , with certain constants θk e μk, 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 ′′‖2 +B‖f‖2 Aθk +Bμk , hold. In each case we determine the corresponding extremal polynomials for which equalities are attained.