Vol. 41, No. 2, 1972

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Inequalities for polynomials with a prescribed zero

John D. Donaldson and Qazi Ibadur Rahman

Vol. 41 (1972), No. 2, 375–378

Let 𝒫n denote the linear space of polynomials p(z) = k=0nakzk of degree at most n. There are various ways in which we can introduce norm ( ) in 𝒫n. Given β let 𝒫n,β denote the subspace consisting of those polynomials which vanish at β. Then how large can p(z)(z β)be if p(z) ∈𝒫n,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)= (12π 02π|p(ei𝜃)|2 d𝜃)12.

Mathematical Subject Classification
Primary: 30A08
Received: 19 January 1971
Published: 1 May 1972
John D. Donaldson
Qazi Ibadur Rahman