Flatness of conjugate reciprocal unimodular polynomials

作者：Erdelyi Tamas^*

来源：Journal of Mathematical Analysis and Applications, 2015, 432(2): 699-714.

DOI：10.1016/j.jmaa.2015.06.020

• Littlewood polynomials
• Unimodular polynomials
• Conjugate reciprocal polynomials
• Flatness properties

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.

• 出版日期2015-12-15