Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients

Abstract

LetP ^c_n,k denote the set of all polynomials of degree at mostn withcomplex coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Let denote the set denote the set of all polynomials of degree at mostn withreal coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Associated with0≤k≤n andx∈[−1, 1], let

$$B_{n,k,x}^* : = \max \{ \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,n\log (\frac{e}{{1 - x^2 }}\} ,B_{n,k,x}^* : = \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,$$

, andM ^*_n,k ≔max{n(k+1),nlogn},M _n,k ≔n(k+1). It is shown that

$$M_{n,k}^* : = \max \{ n(k + 1),n\log n\} ,M_{n,k}^* :n(k + 1)$$

for everyx∈[−1, 1], wherec ₁>0 andc ₂>0 are absolute constants. Here ‖·‖_[−1,1] denotes the supremum norm on [−1,1]. This result should be compared with the inequalities

$$c3\min \{ B_{n,k,x,} B_{n,,k,} \} \leqslant _{p \in P_{n,k} }^{\sup } \frac{{|p'(x)|}}{{||p||[1,1]}} \leqslant \{ B_{n,k,x,} B_{n,,k,} \} ,$$

, for everyx∈[−1,1], wherec ₃>0 andc ₄>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 Erdös in 1940.