AN IMPROVEMENT AND A GENERALIZATION OF ANKENY AND RIVLIN’S RESULT ON THE MAXIMUM MODULUS OF POLYNOMIALS

AN IMPROVEMENT AND A GENERALIZATION OF ANKENY AND RIVLIN’S RESULT ON THE MAXIMUM MODULUS OF POLYNOMIALS

R. Ngamchui, R. Laishangbam, B. Chanam, R. Thoudam

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Abstract

For an arbitrary entire function f(z), let M(f, r) = max |z|=r |f(z)|. By considering the polynomial of degree n having no zero in the interior of the unit circle |z| = 1, Ankeny and Rivlin obtained M(p,R) ≤ Rn + 1 2 M(p, 1), R ≥ 1. In this paper, we consider the polynomial of degree n having no zero in |z| < k, k ≥ 1 and simultaneously considering the sth derivative, 0 ≤ s < n, of the polynomial, we have obtained an improvement as well as a generalization of Ankeny and Rivlin’s result.

Keywords

Polynomial, maximum modulus principle, generalization, sth derivative.