32011 mjis bilal

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DOI: 10.15415/mjis.2015.32011

On The Maximum Modulus of a Polynomial Bilal Dar1, Abdullah Mir2, Q. M. Dawood3, M.I. Shiekh4 and Mir Ahsan Ali5 Department of Mathematics, University of Kashmir, Hazratbal Srinagar-190006 (India). 1,2,3

Department of Mathematics, Bhoj Mahavidyalaya, Bhopal Barkatulla Vishvavidyala, Bhopal, India 4,5

Email: darbilal85@ymail.com Received: 9 August 2014| Revised: 28 February 2015| Accepted: 4 March 2015 Published online: March 30, 2015 The Author(s) 2015. This article is published with open access at www.chitkara.edu.in/publications

Abstract: Let P(z) be a polynomial of degree n not vanishing in |z| < k where k > = 1. It is known that n

(R + k) Max|z|= R>1 | P ( z ) | ≤ n n ( R + k ) + (1 + Rk ) n    n 1 + Rk     n    Min|z|=k | P ( z ) |. ( R + 1) Max|z|=1 | P ( z ) | − R −      R + k    

In this paper, we obtain a refinement of this and many other related results. Key words and Phrases: Polynomial, Zeros, Maximum modulus. Mathematics Subject Classification (2010): 30A10, 30C10, 30D15 1. Introduction and Statement of results For an arbitrary entire function, let M ( f , r ) = Max|z|=r | f ( z ) | m( f , r ) = Min|z|=r | f ( z ) | . Let P(z) be a polynomial of degree n, then

M ( P, R ) ≤ R n M ( P,1), R ≥ 1.

and (1)

Inequality (1) is a simple deduction from Maximum Modulus Principle (see [6], p-442). It was shown by Ankeny and Rivilin [1] that if P(z) does not vanish in |z| < 1, then (1) can be replaced by

M ( P, R ) ≤

Rn + 1 M ( P,1), R ≥ 1. 2

(2)

Mathematical Journal of Interdisciplinary Sciences Vol. 3, No. 2, March 2015 pp. 125–129


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