Markov’s inequality for polynomials with real zeros

Abstract

Markov’s inequality asserts that | | p n ′ | | ⩽ n 2 | | p n | | ||{p’_n}|| \leqslant {n^2}||{p_n}|| for any polynomial p n {p_n} of degree n n . (We denote the supremum norm on [ − 1 , 1 ] [ - 1,1] by | | . | | ||.|| .) In the case that p n {p_n} has all real roots, none of which lie in [ − 1 , 1 ] [ - 1,1] , Erdös has shown that | | p n ′ | | ⩽ e n | | p n | | / 2 ||{p’_n}|| \leqslant en||{p_n}||/2 . We show that if p n {p_n} has n − k n - k real roots, none of which lie in [ − 1 , 1 ] [ - 1,1] , then | | p n ′ ⩽ c n ( k + 1 ) | | p n | | ||{p’_n} \leqslant cn(k + 1)||{p_n}|| , where c c is independent of n n and k k . This extension of Markov’s and Erdös’ inequalities was conjectured by Szabados.