Abstract

The classical Markov inequality says that if a polynomial p of degree n is bounded by 1 on [-1,1], then the maximal value of its k-th derivative p(k) in the same interval is attained by the Chebyshev polynomial T_n(x) = cos n arccos x, i.e., |p(k)(x) | < T_n(k)(1).

In 1941, Duffin and Schaeffer found a refinement of this theorem to the effect that the conclusion still holds when the uniform boundedness |p(x)| < 1 is replaced by a weaker conditions that |p| < 1 only at n+1 points at which |T_n| = 1. Moreover, under such weaker assumption, the Markov inequality can be extended to the complex plane, namely |p(k)(x iy)| < |T_n(k)(1 iy)|.

A crucial role in their proof is played by the so-called end-point domination property of T_n which is |T_n(x iy)| < |T_n(1 iy)|. We show that this property is featured more generally by the ultraspherical polynomials P_n (which Chebyshev polynomials are a part of), and thus Duffin-Schaeffer-type inequalities are valid for them too. The proof is based on an expansion formula for the squared modulus of an entire function from the Laguerre-Polya class, due to Jensen, and some further results, In particular, we prove a conjecture of Patrick from 1971 concerning the Jacobi case.