Walter Bergweiler, Alexandre Eremenko
We consider transcendental entire functions of finite order for which the zeros and $1$-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all.
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Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb{D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb{D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has `minimal' growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup_{z\in \mathbb{D}} (1-|z|)^p |a(z)|=+\infty$ for any $p>0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functio…
Entire periodic functions with plane zeros
给出了具有平面零点的$\C^n$中整周期函数的因子的完整描述。
In this paper, we extend our earlier result (see [Y-2008]) on the distribution of normalized zero-sets of random entire functions to random entire functions with small random perturbation.
Zero sets of some classes of entire functions
给出了一种构造具有给定零点的整函数的方法和增长性估计。它给出了一种可能性来描述零集的某些类的整个功能的一个和几个变量的增长方面的体积，这些集在某些polycylinders。
We give a complete description of zero sets for some well-known subclasses of entire functions of exponential growth (bounded on real axis, Cartwright's class)
We announce conditions under which a given sequence of points on the complex plane is a subsequence of zeros of an entire function with weight restrictions on growth.
We consider transcendental entire solutions of linear $q$-difference equations with polynomial coefficients and determine the asymptotic behavior of their Taylor coefficients. We use this to show that under a suitable hypothesis on the associated Newton-Puiseux diagram their zeros are asymptotic to finitely many geometric progressions. We also sharpen previous results on the growth rate of entire solutions.
It is known that if the proximate order $ρ(r)$ such that $\lim ρ(r) = ρ> 0 (r \to \infty)$, then there exists an entire function $f(z)$ of proximate order $ρ(r)$. In the case where $ρ= 0$ the question about the existence of such entire function remained open until now. This question is investigated in the paper.
We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang circle theorem from statistical mechanics. Also, we state several open problems.
For a real non-signdefinite function $B(z)$, $z\in \C$, we investigate the dimension of the space of entire analytical functions square integrable with weight $e^{\pm 2F}$, where the function $F(z)=F(x_1,x_2)$ satisfies the Poisson equation $\D F=B$. The answer is known for the function $B$ with constant sign. We discuss some classes of non-signdefinite positively homogeneous functions $B$, where both infinite and zero dimension may occur. In the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. The topic is closely r…
On the Zeros of Certain Entire Functions
本研究证明了一类整q-函数存在无穷多非零根且根的模至少指数增长，并将结果应用于含平面波函数q-类似物在内的级数展开定义的整q-函数，丰富了q-特殊函数的零点理论。
On common zeros of entire functions of exponential growth
本文研究 $\mathbb C^n$ 中指数增长整函数方程组的无限公共零点，利用凸集几何刻画其渐近分布，将 Kushnirenko–Bernstein–Khovanskii 型零点计数结果推广到渐近密度情形。
In this paper we shall consider the assymptotic growth of $|P_n(z)|^{1/k_n}$ where $P_n(z)$ is a sequence of entire functions of genus zero. Our results extend a result of J. Muller and A. Yavrian. We shall prove that if the sequence of entire functions has a geometric growth at each point in a set $E$ being non-thin at $\infty$ then it has a geometric growth in $\CC$ also. Moreover, if $E$ has some more properties, a similar result also holds for a more general kind of growth. Even in the case where $P_n$ are polynomials, our results are new in the sense that it does not require $k_n\succeq d…
In the early 1960s, P. Malliavin and L. A. Rubel gave a complete description of pairs of distributions of positive points $Z$ and $W$ such that for each entire function of exponential type $g\neq 0$ that vanishes on $W$, there is an entire function of exponential type $f\neq 0$ such that $f$ vanishes on $Z$ and satisfies the inequality $|f|\leq |g|$ everywhere on the imaginary axis. We extend this result to much more general distributions of complex points $Z$ and $W$ lying outside of some pair of vertical angles containing the imaginary axis as the points approach $\infty$.
This paper is concerned with the distribution of normalized zero-sets of random entire functions. The normalization of the zero-set is performed in the same way as that of the counting function for an entire function in Nevanlinna theory. The result generalizes the Shiffman and Zelditch theory on the distribution of the zeroes of random holomorphic sections of powers for positive Hermitian holomorphic line bundles from polynomial functions to entire functions. Our result can also be viewed as the analogy of Nevanlinna's First Main Theorem in the theory of the distribution of zero-sets of rando…
We study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are constructed, using the Stokes phenomenon for second order linear differential equations.
We study the asymptotics of zeros for entire functions of the form \sin z + \int_{-1}^1 f(t)e^{izt}dt with f belonging to a space X \hookrightarrow L_1(-1,1) possessing some minimal regularity properties.
The paper studies entire functions of finite order of growth for which a representation of the form $ψ(z) = 1+ O(|z|^{-μ}), μ>0,$ as $z\to \infty$, is valid on a fixed ray of the complex plane. The main result is the following. Assume that the zeros of two functions $ψ_1, ψ_2$ of this class coincide in the circle of radius $R$ with the center in zero. Then given arbitrary small $δ\in (0,1)$ and $\varepsilon >0$ the relation of these functions admits the estimate $|ψ_1(z)/ψ_2(z) -1| \leqslant \varepsilon R^{-μ(1-δ)}$ for all $|z|\leqslant R^{1-δ}$, provided that $R\geqslant R_0$ and $R_0 =R_0(\…
If f is a real entire function and ff" has only real zeros then f belongs to the Laguerre-Polya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman in 1914.
The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By using the theorem, we reprove some results about zeros of derivative of Xi function,Gamma function and digamma function in a new way.
Baker's conjecture states that a transcendental entire function of order less than $1/2$ has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits of unbounded wandering domains whenever the order is less than 1. This rai…
Let $P$ and $Q$ be polynomials in one variable over an algebraically closed field $k$ of characteristic zero. Let $f$ and $g$ be elements of a function field $\K$ over $k$ such that $P(f)=Q(g).$ We give conditions on $P$ and $Q$ such that the height of $f$ and $g$ can be effectively bounded, and moreover, we give sufficient conditions on $P$ and $Q$ under which $f$ and $g$ must be constant.
Let $f_ω(z)=\sum\limits_{j=0}^{\infty}χ_j(ω) a_j z^j$ be a random entire function, where $χ_j(ω)$ are independent and identically distributed random variables defined on a probability space $(Ω, \mathcal{F}, μ)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Steinhaus entire functions. Then, we prove that, for almost all functions in the family and for any constant $C>1$, there exist a constant $r_0=r_0(ω)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r>r_0$ and $r\notin E$, $$ |\log M(r, f)- N(r,0, f_…
We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the real axis is deduced. A connection is established to the positive definite functions.
We prove Polya's conjecture of 1943: For a real entire function of order greater than 2, with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity with n. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.
Uniqueness sets with angular density for spaces of entire functions, I. Basics
本文研究了具有角密度的整个函数空间的唯一性集，确定了临界零集类型并给出了其与临界唯一性集类型相吻合的条件，同时探讨了其在福克型空间中的应用。
In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane. We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kern…
If $\sum_n n! c_n z^n$ is entire and $c_n$ does not terminate, then $\sum_n c_n z^n$ has infinitely many zeros
本文证明了整函数系数非终止时其原级数有无穷多零点，并据此为 Le Roy 函数和 Bessel 函数的无穷多零点性质提供了替代证明。
Local repulsion between zeros and critical points of the Gaussian Entire Function
本文研究标准高斯整函数零点与临界点的二阶相关性，给出小圆盘上的渐近结果，揭示不同指标的局部排斥率，并指出这对噪声谱图信号处理中局部极大值与零点强排斥的意义。
We study the hole probability of Gaussian random entire functions. More specifically, we work with the flat model (the zero set of this function has a distribution which is invariant with respect to the plane isometries). A hole is the event where the function has no zeros in a disc of radius r. We show that the logarithm of the probability of the hole event decays asymptotically like -1/4 * e^2 * r^4 + o(r^4). We also study the behavior of the hole probability with other types of random coefficients.
We present a new, short and independent proof of the Liouville-type theorem for entire and subharmonic functions of finite order bounded outside some set of zero planar density.
In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in \cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire functions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have \begin{eqnarray*} \limsup_{n\to\infty}|P_n(z)|^{1/k_n}\leq 1 \end{eqnarray*} for all $z\in \mathbb C$ (see Theorem \ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead natura…
We study Picard's exceptional values of holomorphic one-parametric families of entire functions. Our first result shows that the set of parameter values for which zero is a Picard value can be an arbitrary closed set of zero logarithmic capacity. This answers a question of Julia. Second, we show that if a function has a Picard exceptional value for all values of parameter in some region of the plane then this Picard value is a holomorphic function in the complement of some discrete set E, tending to infinity as the papameter tends to E.
We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval.
In this paper we consider a random entire function of the form $f(z,ω)=\sum\nolimits_{n=0}^{+\infty}ξ_n(ω)a_nz^n,$ where $ξ_n(ω)$ are independent standard\break complex gaussian random variables and $a_n\in\mathbb{C}$ satisfy the relations\break $\varlimsup\limits_{n\to+\infty}\sqrt[n]{|a_n|}=0$ and $ \#\{n\colon a_n\neq0\}=+\infty.$ We investigate asymptotic properties of the probability $P_0(r)=P\{ω\colon f(z,ω)$ has no zeros inside $r\mathbb{D}\}.$ Denote $ p_0(r)=\ln^-P_0(r),\ N(r)=\#\{n\colon \ln (|a_n|r^n)>0\},$ $ s(r)=\sum_{n=0}^{+\infty}\ln^+(|a_n|r^{n}). $ Assuming that $a_0\neq0$ we …
Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions J_α(z) when α\ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^α(z) when α\ge -2, Jacobi polynomials P_n^{(α,β)}(z) when α\ge -1 and β\ge -1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojević and C. V. Stanojević, eds., Marc…
Taylor coefficients and zeroes of entire functions of exponential type
研究全纯指数型函数的泰勒系数与零点分布，证明若零点计数函数增长线性则函数为指数函数，或系数部分为单位模时同样成立，扩展了卡尔松经典结果。
Let $Z$ and $W$ are distributions of points on the complex plane $\mathbb C$. The following problem goes back to the studies of F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane and others. For which $Z$ and $W$ for an entire function $g\neq 0$ of exponential type vanishing on $W$, there is an entire function $f\neq 0$ of exponential type vanishing on $Z$ such that $|f|\leq |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $Z$ and $W$ lying only on the positive semiaxis. A number of …
As an analogue of the Hadamard quotient problem in number theory, the quotient problem (in the sense of complex entire functions) for two sequences $F(n)=a_0+a_1f_1^n+\cdots+a_lf_l^n$ and $ G(n)=b_0+b_1g_1^n+\cdots+b_mg_m^n$, has been solved, where the $f_i$ and $g_j$ are nonconstant entire functions and $a_i$ and $b_j$ are non-zero constants except that $a_0$ can be zero. In this paper, we consider the generalization of this problem in which we allow $a_i$ and $b_j$ to be small growth entire functions with respect to $(g_1, \cdots, g_m)$ by modifying the second main theorem with moving target…
Several years ago, Aziz and Zargar, while considering some questions related to Sendov's conjecture, solved a problem posed by Brown (see [1,2]), showing that any complex polynomial of degree $n$ with a single zero at $z=0$ does not have any critical point in $B(0,1/n)$. More recently, this result has been generalised in [3] by Zargar and Ahmad. The aim of our paper is to extend the result to some classes of complex entire functions. We will show that, under some conditions on the zeros $a_n$ of $f$, $f'$ has no roots in $B(0,t) \setminus \lbrace 0 \rbrace$ for a certain $t$ depending on the v…
For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k,$ $a_k >0,$ we define the sequence of the second quotients of Taylor coefficients $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$. We find new necessary conditions for a function with a non-decreasing sequence $Q$ to belong to the Laguerre--Pólya class of type I. We also estimate the possible number of nonreal zeros for a function with a non-decreasing sequence $Q.$
In this paper we determine a class of entire functions using conditions on their odd and even parts. Further it is shown that the zeros of members of this class are localized in a very special way. This result allows us to treat a particular case of a conjecture by A. Sokal.
For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that if $f$ belongs to the Laguerre-Pólya class, and the quotients $q_k := \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots $ satisfy the condition $q_2 \leq q_3,$ then $f$ has at least one zero in the segment $[-\frac{a_1}{a_2},0].$ We also give necessary conditions and sufficient conditions of the existence of such a zero in terms of the quotients $q_k$ for $k=2,3, 4.$
Abstract. In this work we use an elementary method to derive an upper bound on the right half-plane for genus 0 entire functions if it has only negative zeros. The bound only uses information of the function on the positive real axis. Applications to the modified Bessel functions and the Airy function are also provided.
Let $g$ be a entire function of exponential type on the complex plane $\mathbb C$, $Z=\{ z_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\neq 0$ of exponential type that vanishes on $ Z$ and satisfies the constraint $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$, $y\to \pm\infty$. Our results generalize and develop a joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions on the complex plane, as well as to the completeness…
We study the effect of finite difference operators of finite order on the distribution of zeros of polynomials and entire functions.
Entire functions with an arithmetic sequence of exponents
本文研究整函数按指数模 $m$ 分类所得子级数的零点分布，并给出其零点落在 $m$ 条特定辐射射线上的条件。
Periodicity of Transcendental Entire Functions Sharing Set with their Shifts
本文研究超阶小于1的超越整函数的周期性，证明了当其与平移以权重2共享某集合时函数必具有周期性。
The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results have been obtained. In this paper, we study a transcendental entire function f that shares a non-zero polynomial a with f', together with its linear differential polynomials of the form with rational function coefficients.
We characterize all linear operators which preserve spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip into real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a "strong universal factor" as defined by de Bruijn. Using elementary methods, we prove a theorem …
We apply Rossi's half-plane version of Borel's Theorem to study the zero distribution of linear combinations of $\mathcal{A}$-entire functions (Theorem 1.2). This provides a unified way to study linear $q$-difference, difference and differential operators (with entire coefficients) preserving subsets of $\mathcal{A}$-entire functions, and hence obtain several analogous results for the Hermite-Poulain Theorem to linear finite ($q$-)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomi…
B.\,Ya.\,Levin has proved that zero set of a sine type function can be presented as a union of a finite number of separated sets, that is an important result in the theory of exponential Riesz bases. In the present paper we extend Levin's result to a more general class of entire functions $F(z)$ with zeros in a strip $0<q \leq \Im z \leq Q,$ such that $|F(x)|^2$ satisfies the Helson--Szegö condition. Moreover, we demonstrate that instead of the last condition one can require that $\log|F(x)|$ belongs to the BMO class.
A separated sequence $Λ$ on the real line is called a Polya sequence if any entire function of zero exponential type bounded on $Λ$ is constant. In this paper we solve the problem by Polya and Levinson that asks for a description of Polya sets. We also show that the Polya-Levinson problem is equivalent to a version of the so-called Beurling gap problem on Fourier transforms of measures. The solution is obtained via a recently developed approach based on the use of Toeplitz kernels and de Branges spaces of entire functions.
This monograph is devoted to the theory of entire functions of several variables. A definition of bounded index was supposed by B. Lepson. We generalised his definition for several variables and obtained criteria of L-index boundedness in direction. In particular we obtained obtain analogues of one-dimensional criterion of boundedness L-index in terms of behaviour the logarithmic derivative outside of zero sets, sufficient conditions of L-index boundedness in direction for some subclass of entire functions with "plane" zeros, sufficient conditions of L-index boundedness of entire solutions for…
We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metr…
We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq 2\sqrt[3]{2},$ then all but a finite number of zeros of $f$ are real and simple. We also present a criterion in terms of the closest to zero roots for such a function to have only real zeros (in other words, for belonging to the Laguerre--Pólya class of type I) under additional assumption on the sequence $Q.$
We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. The characterization extends previous results of J. Borcea and the author to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.
Zero entropy on entire Grauert tubes
本文证明具有全 Grauert 管的解析流形其测地流具有零拓扑熵，并由此给出 3-流形的完整分类及若干 4、5-流形的拓扑限制与反例。
In 1987, R. B. Paris uses the analytic function \[\label{main} g(w)=\lim_{n\to\infty}(2φ)^n\biggl(\underbrace{\sqrt{1+\sqrt{1+...\sqrt{1+w}}}}_n-φ\biggr),\ \ \ φ=\frac{1+\sqrt{5}}2, \] to estimate the convergence of nested squares to the golden ratio. The function $g$ is non-entire and, perhaps, can not be expressed in terms of some standard known functions. We show that $f(z):=g^{-1}(z)$ is an entire function satisfying Poincare equality. While $f$ has zeros of various multiplicities, it can be expressed in terms of its simple zeros, forming fractal structures similar to Julia sets.
We study operators of the form X+Y where Y has a finite p-th Schatten norm (p<2), and X is self-adjoint and of Hilbert-Schmidt class. Our study is based on new theorems on zero distribution of entire functions of finite order.
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