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Basic Polynomial Inequalities on Intervals and Circular Arcs

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Abstract

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szegő–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Tamás Erdélyi

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  1. Tamás Erdélyi
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Correspondence to Tamás Erdélyi.

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Communicated by Doron S. Lubinsky.

Dedicated to Peter Lax on the occasion of his 87th birthday.

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Erdélyi, T. Basic Polynomial Inequalities on Intervals and Circular Arcs. Constr Approx 39, 367–384 (2014). https://doi.org/10.1007/s00365-013-9208-0

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