Lax-type Inequalities for Polynomials on Subarcs of the Unit Circle

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 approch to see V.S. Videnskii’s Bernstein-type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, a classical polynomial inequality published first in 1960.