Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients
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Let Pcn,k denote the set of all polynomials of degree at most n with complex coefficients and with at most k (0 k n) zeros in the open unit disk. Let Pn,k denote the set of all polynomials of degree at most n with real coefficients and with at most k (0 k n) zeros in the open unit disk. Associated with 0 k n and cursive Greek chi [-1, 1], let B*n,k,cursive Greek chi := max {n(k + 1)/1 - cursive Greek chi2, nlog (e/1 - cursive Greek chi2)}, Bn,k,cursive Greek chi := n(k + 1)/1 - cursive Greek chi2, and M*n,k := max{n(K + 1), nlogn}, Mn,k := n(k + 1). It is shown that c1 min{B*n,k,cursive Greek chi, M*n,k} suppPcn,k |p(cursive Greek chi)|/p[-1,1] c2 min{B*n,k,cursive Greek chi, M*n,k} for every cursive Greek chi [-1,1], where c1 > 0 and c2 > 0 are absolute constants. Here [-1,1] denotes the supremum norm on [-1,1]. This result should be compared with the inequalities c3 min{Bn,k,cursive Greek chi, Mn,k} suppPn,k |p(cursive Greek chi)|/p[-1,1] c4 min{Bn,k,cursive Greek chi, Mn,k}, for every cursive Greek chi [-1,1], where c3 > 0 and c4 > 0 are absolute constants. The upper bound of this second result is also fairly recent; and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-Bernstein type inequalities are concerned. The lower bound of the second result is proved in this paper. It is the final piece in a long series of papers on this topic by a number of authors starting with Erds in 1940.
