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Markov- and Bernstein-Type Inequalities for Polynomials with Restricted Coefficients

Authors:	Borwein Peter Erdélyi Tamás

Abstract:	The Markov-type inequality $\|\|p\prime \|\|_{0,1]} \leqslant cn\log (n + 1)\|\|p\|\|_{0,1]}$ is proved for all polynomials of degree at most n with coefficients from {-1,0,1} with an absolute constant c. Here ·0,1] denotes the supremum norm on 0,1]. The Bernstein-type inequality $\|p\prime (y)\| \leqslant \frac{c} {{(1 - y)^2 }}\|\|p\|\|_{0,1]} ,y \in 0,1)$ is shown for every polynomial p of the form $p(x) = \sum\limits_{j = m}^n {a_j x^j } ,\|a_m \| = 1,\|a_j \| \leqslant 1,a_{j \in } \mathbb{C}$ The inequality $\|p\prime (y)\| \leqslant \frac{c} {{(1 - y)}}\log \left( {\frac{2} {{1 - y}}} \right)\|\|p\|\|_{0,1]} ,y \in 0,1)$ is also proved for every analytic function p on the open unit disk D that satisfies the growth condition $\|p(0)\| = 1,\|p(z) \leqslant \frac{1} {{1 - \|z\|}},z \in D$

Keywords:	Markov Bernstein inequality restricted coefficients small integer coefficients polynomial
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