Inequalities for polynomials with a prescribed zero

John D. Donaldson and Qazi Ibadur Rahman

Vol. 41 (1972), No. 2, 375–378

DOI: 10.2140/pjm.1972.41.375

Abstract

Let 𝒫_n denote the linear space of polynomials p(z) = ∑ _k=0ⁿa_kz^k of degree at most n. There are various ways in which we can introduce norm (∥ ∥) in 𝒫_n. Given β let 𝒫_n,β denote the subspace consisting of those polynomials which vanish at β. Then how large can ∥p(z)∕(z − β)∥ be if p(z) ∈𝒫_n,β and ∥p(z)∥ = 1? This general question does not seem to have received much attention. Here the problem is investigated when (i) ∥p(z)∥ = max_|z|≤1|p(z)|, (ii) ∥p(z)∥ = (1∕2π ∫ ₀^2π|p(e^i𝜃)|² d𝜃)^1∕2.

Mathematical Subject Classification

Primary: 30A08

Milestones

Received: 19 January 1971

Published: 1 May 1972