An improvement of Cauchy radius for the zeros of a polynomial

Abstract

"For a given polynomial p(z) =a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0} of degree n with complex coefficients, the Cauchy radius r_{0} is a unique positive root of the equation |a_{n}| t^{n}-(|a_{n-1}|t^{n-1}+|a_{n-2}| t^{n-2}+ ... +|a_{1}| t+ |a_{0}|) =0. It refers to a radius of the circular region |z|<= r_{0} in which all the zeros of p(z) lie. The basic aim has been to determine the smallest radius, thereby, minimizing the area of the circular region. In this present paper, we have obtained a result which gives an improvement of the Cauchy radius. Also, we produce an annular region whose center is different from the origin in which the zeros of p(z) lie. Moreover, in many cases, our results give better approximations for estimating the region of polynomial zeros than that obtained from many other well-known results."