On the Growth of Polynomials and Entire Functions of Exponential Type

Abstract

Concerning the growth of a polynomial and its derivative, the following inequalities are well known as Bernstein Inequalities. max(|z|=R) |p(z)| <= max(|z|=1) |p(z)|R^n, for R >= 1, (1) max(|z|=1) |p0(z)| <= max(|z|=1) |p(z)|n, (2) max(|z|=rho) |p(z)| >= max(|z|=1) |p(z)|rho^n, for 0 < rho <= 1. (3) All the above inequalities are best possible and are of great importance both from a theoretical point of view and for applications. The thesis consists of three chapters. In Chapter 1, we provide a brief history of these inequalities and provide the proof of the known fact that all three inequalities above are equivalent in the sense that they can be derived from each other. Also, this chapter contains proof of inequality (1), some of its generalizations, and its sharpening when the polynomial does not have a zero at z = 0. In Chapter 2, we study inequality (1) for polynomials having no zeros in {z : |z| < 1}, and then for polynomials having no zeros inside the circle {z : |z| = K}, K > 0, by providing proofs of several results known in this direction. If p(z) is a polynomial of degree n then, as can be easily verified, the function f(z) = p(eiz) is an entire function of exponential type n, and thus the results for entire functions of exponential type can be considered as generalizations of the corresponding results for polynomials. In Chapter 3 we study the generalizations for entire functions of exponential type of inequality (1) and of some other inequalities studied in Chapter 2. Also in this chapter, we provide a partially different proof of a well known result concerning polynomials having no zeros inside the unit circle. Finally, the proof of a known result that sharpens a well known result of R. P. Boas has been provided.