V.K. Jain: On the location of zeros of a polynomial, p.337-352

Abstract:

Observing that for the zeros of polynomial $p(z) = z^n + \sum_{j=0}^{n-1}a_jz^{j}$, Cauchy's bound

$$\nonumber \vert z\vert < 1 + A, \hspace{.5 in} A = \max_{0 \leq j \leq n-1}\vert a_j\vert$$

does not reflect the fact that for $A \rightarrow 0$, all zeros approach the origin $z=0$, Boese and Luther suggested the proper bound

$$\vert z\vert < R', \nonumber$$

$$\nonumber R' = \left\{ \begin{array}{l} \left\{A ( 1- nA)/(1... ...+ 2((nA -1 )/(n+1))\right\} , A \geq 1/n. \end{array} \right.$$

We have obtained a generalization of Boese and Luther's bound by considering the polynomial

$$\nonumber z^n + a_pz^{n-p} + a_{p+1}z^{n-p-1} + \hdots + a_n, 1 \leq p < n$$

and have also suggested certain related results.

Key Words: Cauchy's bound, angle-independent bound, angle-independent zero free bound.

2000 Mathematics Subject Classification: Primary: 30C15;
Secondary: 30C10.

Download the paper in pdf format here.