Duffin and Schaeffer inequality revisited

Abstract

The classical Markov inequality asserts that the $n$-th Chebyshev polynomial $T_n(x)=\cos n\arccos x$, $x\in [-1,1]$, has the largest $C[-1,1]$-norm of its derivatives within the set of algebraic polynomials of degree at most $n$ whose absolute value in $[-1,1]$ does not exceed one. In 1941 R.J. Duffin and A.C. Schaeffer found a remarkable refinement of Markov inequality, showing that this extremal property of $T_n$ persists in the wider class of polynomials whose modulus is bounded by one at the extreme points of $T_n$ in $[-1,1]$. Their result gives rise to the definition of DS-type inequalities, which are comparison-type theorems of the following nature: inequalities between the absolute values of two polynomials of degree not exceeding $n$ on a given set of $n+1$ points in $[-1,1]$ induce inequalities between the $C[-1,1]$-norms of their derivatives. Here we apply the approach from a 1992 paper of A. Shadrin to prove some DS-type inequalities where Jacobi polynomials are extremal. In particular, we obtain an extension of the result of Duffin and Schaeffer.

Keywords

• Markov inequality
• Duffin–Schaeffer inequality
• Chebyshev polynomials
• Jacobi polynomials
• interlacing of zeros

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