Inequalities for derivatives of polynomials of special type

Bernstein-type inequalities for the derivatives of constrained polynomials

“…By an observation of G. G. Lorentz [9], we have p = wQk, where Qk £ nk and n-k w [x) = £ flj(l -XYX" J. with a11 aj■ > 0. 7=0 We may assume that n -k > 1 ; otherwise, (1) gives Theorem 1.…”

“…The fact that c = A can be chosen was pointed out by M. v. Golitschek and G. G. Lorentz. Now assume indirectly that there exists a -a < y <0 such that (9) \q ( From this and the maximality of Q, we deduce |p'(0)| < cyYYkYl)_max{\p(x)\ (P£Sk(0, 1)),…”

“…Let Sn(z, r) he the family of polynomials from nn which have at most k zeros in the open disk of the complex plane with center z and radius r. A number of papers were written on Markov-and Bernstein-type inequalities for the derivatives of polynomials from Sn(0, 1) in certain special cases. When k = 0, see [5], [6], and [9]; when k is small compared with n, [7] for every p € Sn(l/2, 1/2) with some absolute constant c < 18.…”

Some Inequalities of Algebraic Polynomials with Nonnegative Coefficients

“…In the case a = 0, this result is an analogue of a theorem of Lorentz [3] in the L^ norm. Indeed, that theorem holds for a wider class (Lorentz class) of polynomials, which was studied extensively by Scheick [7]. For some subsets of Lorentz class of polynomials, the extremal problem (1) was discussed by Milovanovic and Petkovic [5] for the Jacobi weight.…”

Extremal problems for Lorentz classes of nonnegative polynomials in 𝐿² metric with Jacobi weight

“…These polynomials (transformed to [0,1]) were introduced by G. G. Lorentz [3] and studied extensively by J. T. Scheick [7]. A subset of the Lorentz class Ln for which PÍ¿~1}(-1) = P,li_1)(l) =0 (» = l,...,m) will be denoted by L^m).…”