In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential pol...In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential polynomial,which extends the related result of Li,and Chen et al..An example is given to show that the hypothesis on the zeros of a(z)is necessary.展开更多
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 th...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.展开更多
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.
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 c...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.展开更多
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,...,...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）｜展开更多
Suppose that function f（z） is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f（z）f（k）（z） takes any non-zero finite complex nu...Suppose that function f（z） is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f（z）f（k）（z） takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f（z）f（k） （z）.展开更多
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 ...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.展开更多
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 ...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.展开更多
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 transcen...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）.展开更多
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 o...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.展开更多
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 r...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.展开更多
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 ...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.展开更多
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 l...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.展开更多
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 converge...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.展开更多
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...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.展开更多
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. ...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.展开更多
To obtain a better Vinogradov's bound, assistant function with two or three zeros and all Dirichlet's characters has been evaluated. Assistant function g(X1,X2,X3,X4)with four zeros and all nonprincipal charac...To obtain a better Vinogradov's bound, assistant function with two or three zeros and all Dirichlet's characters has been evaluated. Assistant function g(X1,X2,X3,X4)with four zeros and all nonprincipal characters but X1X2X3X4 is constructed and evaluated in the paper. And the constant obtained is 8.395 511 841 994.展开更多
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...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.展开更多
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）...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）.展开更多
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 coeff...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）.展开更多
文摘In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential polynomial,which extends the related result of Li,and Chen et al..An example is given to show that the hypothesis on the zeros of a(z)is necessary.
文摘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.
基金This work was supported in part by NSFC(11871187)the Fundamental Research Funds for the Central Universities of China。
文摘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.
文摘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.
文摘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）｜
基金Foundation item： Supported by the National Natural Science Foundation of Education Department of Sichuan Province（2002A031） Supported by the ＂11.5＂ Research and Study Programs of SWUST（06zx2116） Supported by the National Natural Science Foundation of China（10271122）
文摘Suppose that function f（z） is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f（z）f（k）（z） takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f（z）f（k） （z）.
文摘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.
基金Supported by the National Science Foundation of China（11471151） Supported by Program for Liaoning Excellent Talents in University（LR2011031）
Acknowledgment The authors would like to thank the referees for kind suggestions and many useful comments
文摘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.
基金supported by the National Natura Science Foundation of China (11171119)Funding of Tianyuan (11226090)
文摘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）.
基金supported by the NSF of Shandong Province, China (ZR2010AM030)the NNSF of China (11171013 & 11041005)
文摘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.
文摘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.
文摘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.
文摘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.
基金The Project is supported by the National Natural Science Foundation of China and by Natural Science Foundation of Zhejiang Province.
文摘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.
基金supported by National Natural Science Foundation of China (10625107)Program for New Century Excellent Talents in University (04-0490)
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China(10171076)Supported by the Scientific and Technical Committee Foundation of Shanghai(03JC14027)
文摘To obtain a better Vinogradov's bound, assistant function with two or three zeros and all Dirichlet's characters has been evaluated. Assistant function g(X1,X2,X3,X4)with four zeros and all nonprincipal characters but X1X2X3X4 is constructed and evaluated in the paper. And the constant obtained is 8.395 511 841 994.
基金supported by UGC under major research project scheme vide No. MRP-MAJOR-MATH-2013-29143
文摘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.
文摘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）.
文摘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）.
