JournalszaaVol. 15, No. 3pp. 474–758

Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial

  • C. Frappier

    École Polytechnique de Montréal, Canada
  • M.A. Qazi

    École Polytechnique de Montréal, Canada
Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial cover

Abstract

Let Pn\mathcal P_n be the class of all polynomials of degree at most nn. If \| \cdot \| denotes the supremum norm on z=1| z | =1 and Mp(R)=maxx=RP(z)M_p(R) = max_{|x|=R} | P(z) |, then for an arbitrary polynomial P(z)=v=0navzvP(z) = \sum ^n_{v=0} a_v z^v in Pn\mathcal P_n the inequality MP(R)RnPM_P(R) ≤ R^n \| P \| holds, with equality if and only if a0==an1=0a_0 = … = a_{n–1} = 0. Given n,kNn,k \in \mathbb N with 0kn10 ≤ k ≤ n–1, let φn,k(R)\varphi _{n,k} (R) be the largest number such that MP(R)+φn,k(R)akRnP(R1)M_P (R)+ \varphi_{n,k}(R)|a_k| ≤ R^n \|P\| (R ≥ 1) for all PPnP \in \mathcal P_n. Values of φn,k(R)\varphi_{n,k} (R) for k=0k=0 and k=1k = 1 are known since some time. We study the case k2k ≥ 2.

Cite this article

C. Frappier, M.A. Qazi, Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial. Z. Anal. Anwend. 15 (1996), no. 3, pp. 474–758

DOI 10.4171/ZAA/726