Annulus containing all the zeros of a polynomial

Annular Bounds for the Zeros of a Polynomial

“…Theorem 6 of Dalal and Govil [15] can generate infinitely many results, including Theorems 4 and 5, giving annulus containing all the zeros of a polynomial, and over the years, mathematicians have shown the usefulness of their results by comparing their bounds with the existing bounds in the literature by giving some examples and thus showing that their bounds are better in some special cases. 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.…”

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

“…Theorem 1.5 can generate infinitely many results, including Theorem 1.2 to 1.4, giving an annulus containing all the zeros of a polynomial, and over the years, mathematicians were comparing the bounds with the existing bounds in the literature by giving some examples. 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.…”

On comparison of annuli containing all the zeros of a polynomial

“…Theorem 1.5 is capable of generating infinitely many results providing annulus containing all the zeros of a polynomial including Theorems 1.3 and 1.4, and over the years, mathematicians have compared their bounds with the existing bounds in the literature by generating some polynomials and showing that for those polynomials their bound is better than the bound obtained from some of the known results. In this regard, Dalal and Govil [6] by proving following two theorems showed that no-matter what results one obtains as a corollary of Theorem 1.5, one can always generate polynomials for which ones bound is better than the existing ones respectively.…”

“…Since the above theorems of Dalal and Govil [6] imply that the bounds computed by substituting different {A k } n k=1 's in the Theorem 1.5 cannot be in general compared hence a natural question arises if there is a class of polynomials for which the bounds obtained by two different theorems can be compared.…”