Inequalities for Lorentz polynomials

We prove a few interesting inequalities for Lorentz polynomials. A highlight of this paper states that the Markov-type inequality max x∈[−1,1] |f (x)| ≤ n max x∈[−1,1] |f(x)| holds for all polynomials f of degree at most n with real coefficients for which f ′ has all its zeros outside the open unit disk. Equality holds only for f(x) := c((1± x) − 2n−1) with a constant 0 6= c ∈ R. This should be compared with Erdős’s classical result stating that max x∈[−1,1] |f (x)| ≤ n 2 ( n n− 1 ) n−1 max x∈[−1,1] |f(x)| for all polynomials f of degree at most n having all their zeros in R \ (−1, 1).