BASIC POLYNOMIAL INEQUALITIES ON INTERVALS AND CIRCULAR ARCS

Dedicated to Peter Lax on the occasion of his 87th birthday Abstract. We prove the right Lax-type inequality on subarcs of the unit circle of the com-plex 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 approach 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 clas-sical polynomial inequalities published first 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. 1