Some extensions of the Markov inequality for polynomials

Let D denote the unit disc of the complex plane and Pn the class of polynomials of degree at most n with complex coefficients. We prove that max z ∈ ∂D ∣∣∣∣pk(z)− pk(z̄) z − z̄ ∣∣∣∣ ≤ n max 0≤j≤n ∣∣∣∣p(eijπ/n) + p(e−ijπ/n) 2 ∣∣∣∣ , where p0 := p belongs to Pn and for k ≥ 0, pk+1(z) := zpk(z) . We also show how this result contains or sharpens certain classical inequalities for polynomials due to Bernstein, Markov, Duffin and Schaeffer and others. 2000 Mathematical Subject Classification. Primary: 41A17 Introduction Let Pn be the class of polynomials p(z) = n ∑ k=0 ak(p)z of degree at most n with complex coefficients. We define, together with D := {z | |z| < 1}, ‖p‖D := max z∈∂D |p(z)| and ‖p‖[−1,1] := max −1≤x≤1 |p(x)|. The famous inequalities of, respectively, Bernstein and Markov state that for any p ∈ Pn ‖p‖D ≤ n‖p‖D (1) and ‖p‖[−1,1] ≤ n‖p‖[−1,1], (2) while ‖p‖[−1,1] ≤ n max 0≤j≤n ∣∣p(cos(jπ/n))∣∣ (3) is a far reaching extension of (2) obtained by Duffin and Schaeffer [3] in 1941. We refer the reader to the recent book by Rahman and Schmeisser [4] or to the survey paper by Bojanov [1] for historical remarks and generalizations of these inequalities. Let us consider a polynomial p(z) := n ∑ k=0 ak(p)z in Pn and an associated polynomial P (z) := n ∑ k=0 ak(p)Tk(z) where Tk denotes, for each integer k ≥ 0, the k Chebyshev polynomial, i.e., Tk(cos θ) = cos(kθ) for any real number θ. We have P (cos θ) = p(e) + p(e−iθ) 2 and applying (3) to P we obtain the inequality ∣∣∣∣eiθp′(eiθ)− e−iθp′(e−iθ) eiθ − e−iθ ∣∣∣∣ ≤ n max 0≤k≤n ∣∣∣∣p(eikπ/n) + p(e−ikπ/n) 2 ∣∣∣∣ (4) valid for any real θ and equivalent to the Duffin and Schaeffer inequality. Given a non-negative number t and a polynomial p(z) := n ∑ k=0 ak(p)z ∈ Pn we define pt(z) := n ∑ k=0 kak(p)z. Clearly, pt ∈ Pn, p0 = p and pt+1(z) = zpt(z) for t ≥ 0. Our main result is the following Theorem 1 For any integer j ≥ 0 and polynomial p ∈ Pn, ∣∣∣∣pj(eiθ)− pj(e) eiθ − e−iθ ∣∣∣∣ ≤ n max 0≤k≤n ∣∣∣∣p(eikπ/n) + p(e−ikπ/n) 2 ∣∣∣∣ (5) for all real θ. Our proof of Theorem 1 is completely independent of the known proofs of (3). This Theorem 1 therefore contains (3) as a special case (j = 1, compare with (4)). It also follows easily from (5) that |pj−1(e)| ≤ n max 0≤k≤n ∣∣∣∣p(ei(θ+kπ/n)) + p(ei(θ−kπ/n)) 2 ∣∣∣∣ , θ real, (6) for all p ∈ Pn and integer j ≥ 1. It is therefore also clear that our Theorem 1 contains an improvement of Bernstein’s inequality (1). We shall also obtain the following extensions of (3): Theorem 2 Let x ∈ [−1, 1] and n ≥ 1 such that |Tn(x)| ≤ T ′ n(x) n2 or else |Tn(x)| ≤ −T ′ n(x) n2 . Then |p′(x)| ≤ |T ′ n(x)| max 0≤k≤n ∣∣∣∣p(cos(kπ n ))∣∣∣∣.