Annular Bounds for the Zeros of a Polynomial. (original) (raw)
Bounds for the zeros of a polynomial
International Journal of Recent Scientific Research
In this paper we find a bound for all the zeros of a polynomial in terms of its coefficients similar to the bound given by Cauchy's classical theorem.
Bounds for the Zeros of Polynomials
2013
In this paper we find bounds for the zeros of a class of polynomials whose coefficients or their real and imaginary parts are restricted to certain conditions. Our results improve and generalize many known results in this direction.
On the location of the zeros of certain polynomials
Publications de l'Institut Math?matique (Belgrade)
We extend Aziz and Mohammad's result that the zeros, of a polynomial P (z) = n j=0 a j z j , ta j a j−1 > 0, j = 2, 3,. .. , n for certain t (> 0), with moduli greater than t(n − 1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P (z), of degree n, with complex coefficients, does not vanish in the disc |z − ae iα | < a/(2n); a > 0, max |z|=a |P (z)| = |P (ae iα)|, for r < a < 2, r being the greatest positive root of the equation x n − 2x n−1 + 1 = 0, and finally obtained an upper bound, for moduli of all zeros of a polynomial, (better, in many cases, than those obtainable from many other known results).
On annuli containing all the zeros of a polynomial
Mathematical and Computer Modelling, 2010
In this paper, we obtain the annuli that contain all the zeros of the polynomial p(z) = a 0 + a 1 z + a 2 z 2 + • • • + a n z n , where a i 's are complex coefficients and z is a complex variable. Our results sharpen some of the recently obtained results in this direction. Also, we develop a MATLAB code to show that for some polynomials the bounds obtained by our results are considerably sharper than the bounds obtainable from the known results.
Note on the location of zeros of polynomials
2011
In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.
Zero Bounds for a Certain Class of Polynomials
In this paper we give a bound for the zeros of a polynomial with complex coefficients. Our results generalize some known results in addition to giving a way for some new results. Mathematics Subject Classification: 30 C 10, 30 C 15
2015
In this paper we consider the problem of finding the number of zeros of a polynomial in a prescribed region by subjecting the real and imaginary parts of its coefficients to certain restrictions.
Bounds for the Zeros of a Polynomial with Restricted Coefficients
Applied Mathematics, 2012
In this paper we shall obtain some interesting extensions and generalizations of a well-known theorem due to Enestrom and Kakeya according to which all the zeros of a polynomial 1 n n P z a z a z a 0 satisfying the restriction lie in the closed unit disk.
Inequalities for Polynomial Zeros
Survey on Classical Inequalities, 2000
This survey paper is devoted to inequalities for zeros of algebraic polynomials. We consider the various bounds for the moduli of the zeros, some related inequalities, as well as the location of the zeros of a polynomial, with a special emphasis on the zeros in a strip in the complex plane.
On the Location of Zeros of Polynomials
2011
In this paper we obtain certain generalizations and refinements of well known Enestrom – Kakeya Theorem for a polynomial under much less restrictions on its coefficients. Keywords and phrases; zero’s, Bounds, Polynomials. Mathematics, Subject classification (2002): 30C10, 30C15
On the Number of Zeros of a Polynomial in a Region
Fasciculi Mathematici, 2016
In this paper, we impose restrictions on the complex coefficients of a polynomial in order to give bounds concerning the number of zeros in a specific region of the complex plane. Our results generalize and refine a good number of results in this area of research.
On comparison of annuli containing all the zeros of a polynomial
Applicable Analysis and Discrete Mathematics
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
Kragujevac Journal of Mathematics, 2022
If P(z) is a polynomial of degree n, then for a subclass of polynomials, Dalal and Govil [7] compared the bounds, containing all the zeros, for two different results with two different real sequences λk > 0, Pn k=1 λk = 1. In this paper, we prove a more general result, by which one can compare the bounds of two different results with the same sequence of real or complex λk, Pn k=0 ♣λk♣ ≤ 1. A variety of other results have been extended in this direction, which in particular include several known extensions and generalizations of a classical result of Cauchy [4], from this result by a fairly uniform manner.
On Zeros of Polynomials with Restricted Coefficients
Kyungpook mathematical journal, 2015
∑ j=0 ajz j be a polynomial of degree n and Re aj = αj, Im aj = βj. In this paper, we have obtained a zero-free region for polynomials in terms of αj and βj and also obtain the bound for number of zeros that can lie in a prescribed region.