A note on the number of zeros of polynomials in an annulus

Xiangdong Yang; Caifeng Yi; Jin Tu

Displaying similar documents to “A note on the number of zeros of polynomials in an annulus”

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we obtain certain results for the polar derivative of a polynomial $p (z) = c_{n} z^{n} + \sum_{j = μ}^{n} c_{n - j} z^{n - j}$ , $1 \leq μ \leq n$ , having all its zeros on $| z | = k$ , $k \leq 1$ , which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo (2020)

Czechoslovak Mathematical Journal

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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${P_{i}}_{i = 1}^{\infty}$ generated by a three-term recurrence relation $P_{i} (x) + Q_{1} (x) P_{i - 1} (x) + Q_{2} (x) P_{i - 2} (x) = 0$ with the standard initial conditions $P_{0} (x) = 1, P_{- 1} (x) = 0,$ where $Q_{1} (x)$ and $Q_{2} (x)$ are arbitrary real polynomials.

The Lehmer constants of an annulus

Artūras Dubickas, Chris J. Smyth (2001)

Journal de théorie des nombres de Bordeaux

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Let $M (α)$ be the Mahler measure of an algebraic number $α$ , and $V$ be an open subset of $ℂ$ . Then its $L (V)$ is inf $M {(α)}^{1 / deg (α)}$ , the infimum being over all non-zero non-cyclotomic $α$ lying with its conjugates outside $V$ . We evaluate $L (V)$ when $V$ is any annulus centered at $0$ . We do the same for a variant of $L (V)$ , which we call the transfinite Lehmer constant $L_{\infty} (V)$ .Also, we prove the converse to Langevin’s Theorem, which states that $L (V) > 1$ if $V$ contains a point of modulus $1$ . We prove the corresponding result for...

Zeros of solutions of certain higher order linear differential equations

Hong-Yan Xu, Cai-Feng Yi (2010)

Annales Polonici Mathematici

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We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation $f^{(k)} + a_{k - 1} (z) f^{(k - 1)} + \dots + a ₁ (z) f^{'} + D (z) f = 0$ , (1) where $D (z) = Q ₁ (z) e^{P ₁ (z)} + Q ₂ (z) e^{P ₂ (z)} + Q ₃ (z) e^{P ₃ (z)}$ , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), $a_{j} (z)$ (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

The zeros of functions of finite order in $C^{n}$

Lawrence Gruman (1983)

Annales Polonici Mathematici

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Uniqueness results for differential polynomials sharing a set

Soniya Sultana, Pulak Sahoo (2025)

Mathematica Bohemica

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We investigate the uniqueness results of meromorphic functions if differential polynomials of the form ${(Q (f))}^{(k)}$ and ${(Q (g))}^{(k)}$ share a set counting multiplicities or ignoring multiplicities, where $Q$ is a polynomial of one variable. We give suitable conditions on the degree of $Q$ and on the number of zeros and the multiplicities of the zeros of $Q^{'}$ . The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021).

Simple zeros of degree 2 $L$ -functions

Andrew R. Booker (2016)

Journal of the European Mathematical Society

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We prove that the complete $L$ -functions of classical holomorphic newforms have infinitely many simple zeros.

Zeros of a certain class of Gauss hypergeometric polynomials

Addisalem Abathun, Rikard Bøgvad (2018)

Czechoslovak Mathematical Journal

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We prove that as $n \to \infty$ , the zeros of the polynomial $_{2} F_{1} [\begin{matrix} - n, α n + 1 \\ α n + 2 \end{matrix}; z]$ cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999–2001) to the case of a complex parameter $α$ and partially proves a conjecture made by the authors in an earlier work.

Some Results on the Properties of Differential Polynomials Generated by Solutionsof Complex Differential Equations

Zinelâabidine LATREUCH, Benharrat BELAÏDI (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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This paper is devoted to considering the complex oscillation of differential polynomials generated by meromorphic solutions of the differential equation $f^{(k)} + A_{k - 1} (z) f^{(k - 1)} + \dots + A_{1} (z) f^{'} + A_{0} (z) f = 0,$ where $A_{i} (z)$ $(i = 0, 1, \dots, k - 1)$ are meromorphic functions of finite order in the complex plane.

On distance between zeros of solutions of third order differential equations

N. Parhi, S. Panigrahi (2001)

Annales Polonici Mathematici

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The lower bounds of the spacings b-a or a’-a of two consecutive zeros or three consecutive zeros of solutions of third order differential equations of the form y”’ + q(t)y’ + p(t)y = 0 (*) are derived under very general assumptions on p and q. These results are then used to show that $t_{n + 1} - t ₙ \to \infty$ or $t_{n + 2} - t ₙ \to \infty$ as n → ∞ under suitable assumptions on p and q, where ⟨tₙ⟩ is a sequence of zeros of an oscillatory solution of (*). The Opial-type inequalities are used to derive lower bounds of the spacings d-a...

Zeros of functions in $D^{p, α}$ spaces

M. Jevtić (1981)

Matematički Vesnik

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Zero points of quadratic matrix polynomials

Opfer, Gerhard, Janovská, Drahoslava

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Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $𝐩$ , i.e., the terms have the form $𝐀_{j} 𝐗^{j} 𝐁_{j}$ , where all quantities $𝐗, 𝐀_{j}, 𝐁_{j}, j = 0, 1, ..., N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $𝐩$ by a matrix...

A survey of some recent results on Clifford algebras in $ℝ^{4}$

Drahoslava Janovská, Gerhard Opfer (2023)

Applications of Mathematics

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We will study applications of numerical methods in Clifford algebras in $ℝ^{4}$ , in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in $ℝ^{4}$ . In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients...

An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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The main result says in particular that if $t (ζ) : = \sum_{ν = - n} ⁿ c_{ν} e^{i ν ζ}$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.

On zeros of differences of meromorphic functions

Yong Liu, HongXun Yi (2011)

Annales Polonici Mathematici

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Let f be a transcendental meromorphic function and $g (z) = f (z + c ₁) + \dots + f (z + c_{k}) - k f (z)$ and $g_{k} (z) = f (z + c ₁) \dots f (z + c_{k}) - f^{k} (z)$ . A number of results are obtained concerning the exponents of convergence of the zeros of g(z), $g_{k} (z)$ , g(z)/f(z), and $g_{k} (z) / f^{k} (z)$ .

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou (2017)

Czechoslovak Mathematical Journal

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Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $∥ p^{'} ∥_{[- 1, 1]} \leq \frac{1}{2} {∥ p ∥}_{[- 1, 1]}$ for a constrained polynomial $p$ of degree at most $n$ , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(- 1, 1)$ and establish a new asymptotically sharp inequality. ...

Sharp estimation of the coefficients of bounded univalent functions close to identity

Lucjan Siewierski

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CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....

Linear maps preserving elements annihilated by the polynomial $X Y - Y X^{†}$

Jianlian Cui, Jinchuan Hou (2006)

Studia Mathematica

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Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by $T^{†}$ the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies $Φ (A) Φ (B) = Φ (B) Φ {(A)}^{†}$ for all A, B ∈ ℬ(H) with $A B = B A^{†}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $Φ (A) = c U A U^{†}$ for all A ∈ ℬ(H).

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let $κ_{p} (n, L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P (x) = \sum_{j = 0}^{n} a_{j} x^{j}$ , $| a_{0} | \geq L (\sum_{j = 1}^{n} | a_{j} |^{p}$ 1/p $,$ aj ∈ ℂ $,$ such that ${(x - 1)}^{k}$ divides P(x). For n ∈ ℕ and L > 0 let $κ_{\infty} (n, L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P (x) = \sum_{j = 0}^{n} a_{j} x^{j}$ , $| a_{0} | \geq L m a x_{1 \leq j \leq n} | a_{j} |$ , $a_{j} \in ℂ$ , such that ${(x - 1)}^{k}$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_{1} \sqrt (n / L) - 1 \leq κ_{\infty} (n, L) \leq c_{2} \sqrt (n / L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Real zeros of general $L$ -functions

Alberto Perelli, Giuseppe Puglisi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni $L(s,\chi)$ , definite in [8], Combinando le tecniche elementari di Pintz [9] con alcuni metodi analitici si ottiene l’estensione dei classici teoremi di Hecke e Siegel.

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .