Electronic Theses and Dissertations

# Geometry of Zeros and Bernstein Type Inequalities concerning Growth for Polynomials

2015-12-08

Nwaeze, Eze R.

Dissertation

## Department

Mathematics and Statistics

In this dissertation, we study the problems related to geometry of zeros and Bernstein type inequalities concerning growth for polynomials. The dissertation consists of three chapters followed by list of references used in the text. In Chapter one, besides presenting a brief introduction of the subject of geometry of zeros of polynomials, we mentioned briefly how the subject of location of zeros of polynomials can be useful in the study of problems in a dynamical system. Let $p(z)=\displaystyle{\sum_{j=0}^{n}a_jz^j}$ be a polynomial of degree $n$ with real coefficients. The well known Enestr\"om-Kakeya theorem states that if $0<a_0\leq a_1\leq a_2\leq a_3\ldots\leq a_n,$ then all the zeros of $p(z)$ lie in the closed unit disk. Here, we generalize and extend this theorem. Also, we derive conditions on coefficients of $p(z)$ and estimate the number of zeros that the polynomial has in a prescribed region. In Chapter two, we obtain several results which provide annuli containing all the zeros of a complex polynomial. Our result are explicit and the radii obtained are in terms of the coefficients of the polynomial. Also, we develop MATLAB code to construct examples of polynomials for which our results give sharper bound than obtainable from some well known results. The problems of this type were initiated by Gauss and Cauchy. In addition, we considered polynomial of the type $a_nz^n+ a_mz^m+a_2z^2+a_1z+a_0$, ~$3\leq m<n$ and obtained a disk centered at the origin that has at least one zeros of the polynomial. Problems of this type were initiated by Landau. If $p(z)=\displaystyle{\sum_{j=0}^{n}a_jz^j}$ is a polynomial of degree $n,$ then it was proved by Bernstein that $\displaystyle\max_{|z|=1}|p'(z)|\leq n\max_{|z|=1}|p(z)|,$ and for $R\geq 1,$~ $\displaystyle\max_{|z|=R}|p(z)|\leq R^n\max_{|z|=1}|p(z)|.$ If $p(z)\neq 0$ in $|z|\leq 1,$ then $\displaystyle\max_{|z|=1}|p'(z)|\leq \frac{n}{2}\max_{|z|=1}|p(z)|$ and $\displaystyle\max_{|z|=R\geq 1}|p(z)|\leq \frac{R^n+1}{2}\max_{|z|=1}|p(z)|.$ The first result was conjectured by Erd$\ddot{o}$s and proved by P. D. Lax and the second result is due to Ankeny and Rivlin. If $0<r<1,$ then Rivlin proved that $\displaystyle\max_{|z|=r}|p(z)|\geq \big(\frac{1+r}{2}\big)^n\max_{|z|=1}|p(z)|.$ Chapter three deals with results in this direction where we prove a refinement of this result of Rivlin. Our result is best possible and gives a sharper result for all polynomials of this class except for polynomials with $\displaystyle\min_{|z|=1}|p(z)|=0.$ This chapter also contains some other results in this direction.