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A well known result due to Ankeny and Rivlin states that if P(z)= a z is a =0 polynomial of degree n with no zeros in z < 1 , then for R ≥ 1 , R + 1 max P(z) ≤ max P(z). z=R 2 z=1 In this paper we obtain some new results which provide several refinements and generalizations of the above inequality and other related results. Keywords Polynomial · Maximum modulus · Zeros Mathematics Subject Classification 30A10 · 30C10 · 30C15 1 Introduction and statement of results For an arbitrary entire function f(z), let M(f , r) ∶= max f (z) and z=r m(f , r) ∶= minf (z) . For a polynomial P(z) of degree n, it is known that z=r M(P, R) ≤ R M(P,1), R ≥ 1. (1.1) Inequality (1.1) is a simple consequence of maximum modulus principle (see [5]). It was shown by Ankeny and Rivlin [1] that if P(z) ≠ 0 in z < 1 , then (1.1) can be replaced by R + 1 M(P, R) ≤ M(P,1), R ≥ 1. (1.2) * Abdullah Mir drabmir@yahoo.com Department of Mathematics, University of Kashmir, Srinagar 190006, India Vol.:(0123456789) 1 3 A. Mir M(P,R) It was proposed by late R. P. Boas to
The Journal of Analysis – Springer Journals
Published: Nov 27, 2018
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