Inequalities between height and deviation of polynomials

Abstract

Abstract In this paper, for polynomials with real coefficients P , Q P,Q satisfying ∣ P ( x ) ∣ ≤ ∣ Q ( x ) ∣ | P\\left(x)| \\le | Q\\left(x)| for each x x in a real interval I I , we prove the bound L ( P ) ≤ c L ( Q ) L\\left(P)\\le cL\\left(Q) between the lengths of P P and Q Q with a constant c c , which is exponential in the degree d d of P P . An example showing that the constant c c in this bound should be at least exponential in d d is also given. Similar inequalities are obtained for the heights of P P and Q Q when the interval I I is infinite and P , Q P,Q are both of degree d d . In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials.