Majorization of the Critical Points of a Polynomial by Its Zeros

Journal of Interdisciplinary Mathematics, 1998

Let f (z) be an arbitrary entire function and M (f, r) = max |z|=r |f (z)|. A wellknown theorem of Ankeny and Rivlin [1] states that if p(z) is a polynomial of degree n satisfying p(z) = 0 for |z| < 1, then for R ≥ 1 max |z|=R |p(z)| ≤ R n + 1 2 max |z|=1 |p(z)|. In this paper we consider the class of polynomials of the form p(z) = a 0 + n ν=μ a ν z ν , 1 ≤ µ ≤ n, having no zeros in |z| < k, k ≥ 1, and obtain results which generalize as well as improve upon the above inequality and some other results known in this direction.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2003

Let f : C n ! C be a polynomial of degree d. We prove that the set of asymptotic critical values of f (i.e. values for which Malgrange's Condition fails) has at most d nÀ1 À 1 points. We give an asymptotically sharp bound for the number of bifurcation points of f. We give also an algorithm to compute this set.

2012

Let P(z) be a polynomial of degree n with real or complex coefficients . The aim of this paper is to obtain a ring shaped region containing all the zeros of P(z). Our results not only generalize some known results but also a variety of interesting results can be deduced from them.

Linear Algebra and its Applications, 2012

We apply several matrix inequalities to the derivative companion matrices of complex polynomials to establish new bounds and majorization relations for the critical points of these polynomials in terms of their zeros.

Illinois Journal of Mathematics, 1997

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics

In this paper we impose distinct restrictions on the moduli of the zeros of p(z)=n∑v=0avzvp(z)=∑v=0navzv and investigate the dependence of ∥p(Rz)−p(σz)∥‖p(Rz)−p(σz)‖, R&gt;σ≥1R&gt;σ≥1 on MαMα and Mα+πMα+π, where Mα=max1≤k≤n|p(ei(α+2kπ)/n)|Mα=max1≤k≤n|p(ei(α+2kπ)/n)| and on certain coefficients of p(z)p(z). This paper comprises several results, which in particular yields some classical polynomial inequalities as special cases. Moreover, the problem of estimating p(1−wn)p(1−wn), $0&lt;w\leq$ given $p(1)=0$ is considered.

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.

2013

The problem of finding the number of zeros of a polynomial in a given circle is of great interest in the theory of the distribution of zeros of polynomials. In this paper we consider the said problem under certain conditions on the coefficients of the polynomial or their real and imaginary parts and prove certain results that generalize some well-known results on the subject.

Let f be a monic polynomial of degree n with zeros z1,. .. , zn. We show that k ν=1 φ |zν | ≤ φ M0(f) + (k − 1)φ(1) (k = 1,. .. , n−1), n ν=1 φ |zν | ≤ φ M0(f) + (n − 2)φ(1) + φ |f (0)| /M0(f) , where φ is any non-decreasing function such that φ • exp is convex on R and M0(f) is the so-called Mahler measure of f. These inequalities describe a weak majorization. Certain upper bounds for the Mahler measure allow us to establish more explicit results, which are still sharp.

Pacific Journal of Mathematics, 1980