Inequalities for Lorentz polynomials

We prove a few interesting inequalities for Lorentz polynomials. A highlight of this paper states that the Markov-type inequality maxx∈[−1,1]|f′(x)|≤nmaxx∈[−1,1]|f(x)| holds for all polynomials ff of degree at most nn with real coefficients for which f′f′ has all its zeros outside the open unit disk. Equality holds only for f(x):=c((1±x)n−2n−1)f(x):=c((1±x)n−2n−1) with a constant 0≠c∈R0≠c∈R. This should be compared with Erdős’s classical result stating that maxx∈[−1,1]|f′(x)|≤n2(nn−1)n−1maxx∈[−1,1]|f(x)| for all polynomials ff of degree at most nn having all their zeros in R∖(−1,1)R∖(−1,1).