Markov-Bernstein type inequalities under Littlewood-type coefficient constraints (Q5935894)

In this paper the authors establish the right Markov-type inequalities on the interval \([0,1]\) for the following classes of polynomials: \(\mathcal F_n=\{p(x)=\sum_{j=0}^na_jx^j:\;a_j\in\{-1,0,1\}\}\), \(\mathcal G_n=\{p(x)=\sum_{j=m}^na_jx^j,\;|a_m|=1,\;|a_j|\leq 1\}\), \(\mathcal L_n=\{p(x)=\sum_{j=0}^na_jx^j:\;|a_j|=1\}\) and \(\{p\in\bigcup_{n=0}^\infty\mathcal F_n:\;|p(0)|=1\}.\) In particular, they show that there are absolute constants \(C_1>0\) and \(C_2>0\) such that NEWLINE\[NEWLINEC_1n\log (n+1)\leq\max_{0\neq p\in\mathcal F_n}\frac{|p^\prime(1)|}{\|p\|_{[0,1]}}\leq\max_{0\neq p\in\mathcal F_n}\frac{\|p^\prime\|_{[0,1]}}{\|p\|_{[0,1]}}\leq C_2n\log(n+1)NEWLINE\]NEWLINE and NEWLINE\[NEWLINEC_1n^{3/2}\leq\max_{0\neq p\in\mathcal G_n}\frac{|p^\prime(1)|}{\|p\|_{[0,1]}}\leq\max_{0\neq p\in\mathcal G_n}\frac{\|p^\prime\|_{[0,1]}}{\|p\|_{[0,1]}}\leq C_2n^{3/2}.NEWLINE\]NEWLINE One of the notable features is that theses bounds are quite distinct from those for unrestricted polynomials.