Inequalities for Sums and Products of Complex Zeros of Solutions to ODE with Polynomial Right Parts

The paper is devoted to non-homogeneous second-order differential equations with polynomial right parts and polynomial coefficients.We derive estimates for the partial sums and products of the zeros of solutions to the considered equations.These estimates give us bounds for the function counting the zeros of solutions and information about the zero-free domains.

PROPORTION OF DISTINCT ZEROS OF A DIRICHLET L-SERIES

By studying the distribution of zeros of combinations of a Dirichlet L-function and its first-order derivative,we prove that every Dirichlet L-function has more than 66.7934%distinct zeros.

ON THE ZEROS OF A CLASS OF POLYNOMIALS AND RELATED ANALYTIC FUNCTIONS

In this paper we prove some interesting extensions and generalizations of Enestrom- Kakeya Theorem concerning the location of the zeros of a polynomial in a complex plane. We also obtain some zero-free regions for a class of related analytic functions. Our results not only contain some known results as a special case but also a variety of interesting results can be deduced in a unified way by various choices of the parameters.

PERTURBATIONS OF ZEROS OF SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH POLYNOMIAL COEFFICIENTS

Let P（z） and P（z） be polynomials of the same degree. We consider the equations u＂ = P（z）u and u＂ = P（z）u （z ∈ C） whose solutions are u（z） and u（z）, respectively. Let Zk（U） and zk（u）, k = 1, 2,..., be the zeros of u（z） and u（z）, respectively. We derive bounds for the quantity sup j inf k｜1/zk（u）-1/zj（u）｜

DISTRIBUTION OF ZEROS OF SOLUTIONS OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

The paper gives two estimates of the distance between adjacent zeros of solutions of the first\|order delay differential equation x′(t)+p(t)x(t-τ) =0 in the case when p(t)≥0 and ∫ t t-τ p(s)d s-1e oscillates or p(t) itself oscillates.

Zeros of the Jones Polynomial for Torus Knots and 2-bridge Knots

We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, ···,(-1)r2) by the recursive form and discuss the distribution of their zeros.

ON ZEROS OF SOLUTIONS OF HIGHER ORDER HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS

In this article, we investigate the exponent of convergence of zeros of solutions for some higher-order homogeneous linear differential equation, and prove that if Ak-1 is the dominant coefficient, then every transcendental solution f（z） of equation……satisfies ）λ（f） =∞, where A（f） denotes the exponent of convergence of zeros of the meromor- phic function f（z）.

ZEROS AND FIXED POINTS OF DIFFERENCE OPERATORS OF MEROMORPHIC FUNCTIONS

Let f be a transcendental meromorphic function and △f（z） = f（z ＋ 1） -- f（z） A number of results are proved concerning the existences of zeros and fixed points of Af（z） and △f（z）/f（z） when f（z） is of order σ（f）=1. Examples show that some of the results are sharp.

Bounds for the Zeros of a Polynomial with Restricted Coefficients

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 P(Z =αnZn+...+α1Z+α0satisfying the restriction αn≥αn-1≥...≥α1≥α0≥0 lie in the closed unit disk.

Estimated Bounds for Zeros of Polynomials from Traces of Graeffe Matrices

In this paper, we combine Graeffe matrices with the classical numerical method of Dandelin-Graeffe to estimate bounds for the moduli of the zeros of polynomials. Furthermore, we give some examples showing significant gain for the convergence towards the polynomials dominant zeros moduli.

OVERCONVERGENCE AND ZEROS OF SEQUENCES OF RATIONAL FUNCTIONS

In the present paper we consider sequences of rational functions with a bounded number or free poles con- verging uniformly with a maximum geometric rate to a holomorphic function on a regular set,and we examine the limiting distribution of the Zeros and its relations with the phenomenon of overconvergence.Our results further extend the well known classical theory of overconvergence and the zeros of sections of Taylor series.

A SIMPLE ITERATIVE PROCEDURE FOR FINDING ALL ZEROS OF A POLYNOMIAL WITHOUT REPEATED EVALUATION

In this note a simple iterative method for simultaneously finding all zeros of a polynomial is established.The method does not require repeated evaluation of the polynomial or its deriva-tives,and is globally convergent for quadratic polynomials.

CONTINUITY AND HAUSDORFF DIMENSION OF JULIA SET CONCERNING YANG-LEE ZEROS

Considering the Julia set J（Tλ） of the Yang-Lee zeros of the Potts model on the diamond hierarchical Lattice on the complex plane, the authors proved that HDJ（Tλ） 〉 1 and discussed the continuity of J（Tλ） in Hausdorff topology for λ∈R.

Some Inequalities for the Polynomial with S-Fold Zeros at the Origin

Let p（z） be a polynomial of degree n, which has no zeros in ｜z｜〈 1, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 （2010）, pp. 38-41] established｜zp＇（z）＋nβ/2p（z）｜≤n/2{（｜β/2｜＋｜1＋β/2｜）｜z｜=1max｜p（z）｜-（｜1＋β/2｜-｜β/2｜）｜z｜=1min｜p（z）｜},for any ｜β｜≤ 1 and ｜z｜ = 1. In this paper we improve the above inequality for the polynomial which has no zeros in ｜z｜ 〈 k, k≥ 1, except s-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.

Some Integral Mean Estimates for Polynomials with Restricted Zeros

Let P（z） be a polynomial of degree n having all its zeros in ｜z｜≤ k. For k = 1, it is known that for each r 〉 0 and ｜a｜≥1,n（｜a｜-1）{∫0^2π｜P（e^iθ）｜^rdθ}^1/r≤{∫0^2π｜1＋e^iθ｜^rdθ}^1/rmax｜z｜=1｜DzP（z）｜.In this paper, we shall first consider the case when k≥1 and present certain generaliza- tions of this inequality. Also for k≤ 1, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.

On Growth of Polynomials with Restricted Zeros

Let P（z） be a polynomial of degree n which does not vanish in ｜z｜ 〈k, k≥ 1. It is known that for each 0 〈 s 〈 n and 1 ≤ R ≤ k,M（P（s）,R）≤（1/（R^s＋k^s）[{d（s）/dx（s）（1＋x^n）}x=1]（（R＋k）/（1＋k）^nM（P,1）. In this paper, we obtain certain extensions and refinements of this inequality by in- volving binomial coefficients and some of the coefficients of the polynomial P（z）.

ON THE COEFFICIENTS OF A POLYNOMIAL WITH RESTRICTED ZEROS

Let P（z）=anz^n＋an-1z^n-1＋…＋a0be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P（z）. In this paper we prove some sharp inequalities concerning the coefficients of the polynomial P（z） with restricted zeros. We also establish a sufficient condition for the separation of zeros of P（z）.

APPROXIMATION PROPERTIES OF LAGRANGE INTERPOLATION POLYNOMIAL BASED ON THE ZEROS OF (1-x^2)cosnarccosx

We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, x）‖ which reflects the influence of the position of the x＇s and ω（f^（r＋1）,δ）j,j = 0, 1,... , s,on the error of approximation.

The Zeros of a Certain Homogeneous Difference Polynomials of Meromorphic Functions

Let f(z) be a function transcendental and meromorphic in the plane of growth order less than 1. This paper focuses on discuss and estimate the number of the zeros of a certain homogeneous difference polynomials of degree k in f(z), and obtains that this certain homogeneous difference polynomials has infinitely many zeros.

On the Zeros of a Polynomial

In this paper we consider the problem of finding the estimate of maximum number of zeros in a prescribed region and the results which we obtain generalizes and improves upon some well known results.