33 research outputs found
Inequalities for Lorentz polynomials
We prove a few interesting inequalities for Lorentz polynomials including
Nikolskii-type inequalities. A highlight of the paper is a sharp Markov-type
inequality for polynomials of degree at most n with real coefficients and with
derivative not vanishing in the open unit disk. The result may be compared with
Erdos's classical Markov-type inequality (1940) for polynomials of degree at
most n having only real zeros outside the interval (-1,1)
The Mahler measure of the Rudin-Shapiro polynomials
Littlewood polynomials are polynomials with each of their coefficients in
{-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable
flatness property on the unit circle of the complex plane is given by the
Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure
and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of
the complex plane have the same size. It is also shown that the Mahler measure
and the maximum norm of the Rudin-Shapiro polynomials have the same size even
on not too small subarcs of the unit circle of the complex plane. Not even
nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials
have been known before
Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1
Non UBCUnreviewedAuthor affiliation: Texas A & M UniversityFacult