Markov-type inequalities for constrained polynomials with complex coefficients (original) (raw)

1998, Illinois Journal of Mathematics

It is shown that c 1 n max{k + 1, log n} ≤ sup 0 =p∈P c n,k p ′ [−1,1] p [−1,1] ≤ c 2 n max{k + 1, log n} with absolute constants c 1 > 0 and c 2 > 0, where P c 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 c 3 n(k + 1) ≤ sup 0 =p∈P n,k p ′ [−1,1] p [−1,1] ≤ c 4 n(k + 1) , where c 3 > 0 and c 4 > 0 are absolute constants and P n,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 c 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 c 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.