Markov- and Bernstein-type inequalities for polynomials with restricted coefficients
Let \(\| \cdot\| _{[a,b]}\) denote the supremum norm on \([a,b]\). The authors are interested in improvements to the classical Markov and Bernstein inequalities for polynomials whose coefficients are restricted by inequality constraints. For example, they show that if \(0 \leq a < b < 1\), then there is a constant \(c = c(a,b)\) such that if \(p\) is any polynomial of the form \(p(x) = \sum_{j=m}^n a_j x^j\), with \(| a_m| =1\) and \(| a_j| \leq 1\) then \(\| p'\| _{[a,b]} \leq cn\| p\| _{[a,b]}\). They show that there is an absolute constant \(c\) such that if \(p(x)\) is any polynomial of degree \(n\) with all coefficients lying in the set \(\{-1,0,1\}\) then \(\| p'\| _{[0,1]} \leq cn \log(n+1)\| p\| _{[0,1]}\). For the class of polynomials of the previous sentence they show that Bernstein's inequality can be improved to the following remarkable inequality: \(| p'(y)| \leq c(1-y)^{-2} \| p\| _{[0,1]}\) where \(c\) is (another) absolute constant. The proofs make extensive and clever use of Hadamard's three circles theorem.
scientific article; zbMATH DE number 1185670