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 ) n - 2 n - 1 ) with a constant 0 ? c ? R . This should be compared with Erd?s'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 ) .

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