Markov-type inequalities for constrained polynomials with complex coefficients

Abstract

It is shown that c1nmax{k + 1, logn} ≤ sup 06=p∈Pc n,k ‖p‖[−1,1] ‖p‖[−1,1] ≤ c2nmax{k + 1, logn} with absolute constants c1 > 0 and c2 > 0, where P n,k denotes 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. Here ‖ · ‖[−1,1] denotes the supremum norm on [−1, 1]. This result should be compared with the inequalities c3n(k + 1) ≤ sup 06=p∈Pn,k ‖p‖[−1,1] ‖p‖[−1,1] ≤ c4n(k + 1) , where c3 > 0 and c4 > 0 are absolute constants and Pn,k denotes 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. This second result has been known for a few years, and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-type inequalities are concerned. Let Pn(r) denote the set of all polynomials of degree at most n with real coefficients and with no zeros in the union of open disks with diameters [−1,−1+2r] and [1− 2r, 1], respectively (0 < r ≤ 1). Let P n(r) denote the set of all polynomials of degree at most n with complex coefficients and with no zeros in the union of open disks with diameters [−1,−1+2r] and [1− 2r, 1], respectively (0 < r ≤ 1). An essentially sharp Markov-type inequality for P n(r) on [−1, 1] is also established that should be compared with the analogous result for Pn(r) proved in an earlier paper. 1991 Mathematics Subject Classification. 41A17.