A General Theorem on the Bounds of the Derivative of a Polynomial PDF

Keywords

bounds, derivative, polynomial

Abstract

Let ∆ be the Unit disk |z|≤1 in the complex plane C. The well known Bernstein's theorem on the bounds of the derivative of an nth degree polynomial f: C ---> C states that if f(∆) C ∆ then |f'(∆)|≤ n (i.e. |f'(z)|≤ n for |z|≤1). This result was generalized by Szego and sharpened by Lax under an additional condition. Here, we obtain quite a general theorem that deduces all these results as corollaries and· furnishes a chain of interesting new results, some of which offer more general versions (sometimes sharper estimates for |f'(z)|) of the theorems of Bernstein, Szego, and Lax. In fact, we present a unified approach to the basic nature of the problem and its solution underlying Bernstein's theorem and other related Bernstein-type results.