Flatness of conjugate reciprocal unimodular polynomials

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:432

Article

Erdelyi, Tamas¹ 

[1] Texas A&M Univ, Dept Math, College Stn, TX 77843 USA

关键词: Littlewood polynomials;    Unimodular polynomials;    Conjugate reciprocal polynomials;    Flatness properties;   

DOI  :  10.1016/j.jmaa.2015.06.020

来源: Elsevier

【 摘 要 】

A polynomial is called unimodular if each of its coefficients is a complex number of modulus 1. A polynomial P of the form P(z) = Sigma(n)(j=0) a(j)z(j) is called conjugate reciprocal if a(n-j) = (a) over bar (j), a(j) is an element of C for each j = 0,1, ... , n. Let partial derivative D be the unit circle of the complex plane. We prove that there is an absolute constant epsilon > 0 such that max(z is an element of partial derivative D) vertical bar f(z)vertical bar >= (1 + epsilon)root 4/3m(1/2), for every conjugate reciprocal unimodular polynomial of degree m. We also prove that there is an absolute constant epsilon > 0 such that M-q(f') <= exp(epsilon(q - 2)/q)root 1/3m(3/2), 1 <= q < 2, and M-q(f') >= exp(epsilon(q - 2)/q)root 1/3m(3/2), 2 < q, for every conjugate reciprocal unimodular polynomial of degree m, where M-q(g) = (1/2 pi integral(2 pi)(0) vertical bar g(e(it))vertical bar(q) dt)(1/q), q > 0. (C) 2015 Elsevier Inc. All rights reserved.

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