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## On the Growth of Polynomials and Entire Functions of Exponential Type

##### Date

2004-12-15##### Author

Harden, Lisa

##### Type of Degree

Thesis##### Department

Mathematics##### Metadata

Show full item record##### 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.