Abstract
The Bernstein inequality states that if p(z) is a polynomial of degree at most n, then ‖p′‖⩽n‖p‖, where ‖p‖=max|z|=1|p(z)|. The Lr norm of polynomial p is‖p‖r=(12π∫02π|p(eiθ)|rdθ)1/r for r⩾1 (we then have limr→+∞‖p‖r=‖p‖∞=‖p‖). This chapter considers Bernstein-type inequalities in terms of the Lr norm. Results are also presented for 0⩽r<1. Refinements are given for polynomials with restrictions on the location of their zeros. Results comparing Lr1 norms of p′ to Lr2 norms of p are given where proofs involve Hölder's inequality. Refinements involving min|z|=K|p(z)| are given and results concerning higher-order derivatives are addressed. Besides, inequalities for self-inversive, as well as for some other classes of polynomials, are investigated.
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