On comparison of annuli containing all the zeros of a polynomial

Abstract

There are many theorems providing annulus containing all the zeros of a polynomial, and it is known that two such theorems cannot be compared, in the sense that one can always find a polynomial for which one theorem gives a sharper bound than the other. It is natural to ask if there is a class of polynomials for which such comparison is possible and in this paper we investigate this problem and provide results which for polynomials with some condition on the degree or absolute range of coefficients, enable us to compare two such theorems.

A Note on Comparison of Annuli Containing all the Zeros of a Polynomial

“…Dalal and Govil [6] have shown that these bounds cannot, in general, be compared, implying that every result obtained can be useful. More recently, Dalal and Govil [7] successfully compared the bounds of two different results with two different real sequences λ k > 0, n k=0 λ k = 1 for a subclass of polynomials. The main aim of this paper is to prove the following more general results (Theorem 2.1, 2.2) of Section 2 using well known Hölder Inequality…”

Annular Bounds for the Zeros of a Polynomial

“…In this connection, Dalal and Govil in [16] have shown that no matter what result you obtain as a corollary to Theorem 6, one can always generate polynomials for which the corollary so obtained gives better bound than the existing ones, implying that every result obtained by Theorem 6 can be useful. Since the results obtained as corollaries of Theorem 6 cannot in general be compared, more recently Dalal and Govil [17] have given results that help to compare the bounds for a subclass of polynomials. For this, they provide a class of polynomials with some conditions on degree or absolute range of coefficients of the polynomial, and for this class of polynomials, the bound obtained by one corollary is always better than the bound obtained from the other.…”