Duffin and Schaeffer inequality revisited

Year 2025, Volume: 8 Issue: 2, 81 - 92, 15.06.2025

Geno Nikolov

https://doi.org/10.33205/cma.1632536

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

Supporting Institution

European Union-NextGenerationEU, through the National Recovery and Resilience Plan of the Republic of Bulgaria

Project Number

project No BG-RRP-2.004-0008

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