Walter Bergweiler, A. Eremenko, J. K. Langley
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.
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Generalizations of the theorems of Wiman and of Arima on entire functions are proved for spatial quasiregular mappings.
We establish a Wiman-Valiron theory for a polynomial series based on the Wilson operator $\mathcal{D}_\mathrm{W}$. For an entire function $f$ of order smaller than $\frac13$, this theory includes (i) an estimate which shows that $f$ behaves locally like a polynomial consisting of the terms near the maximal term in its Wilson series expansion, and (ii) an estimate of $\mathcal{D}_\mathrm{W}^n f$ compared to $f$. We then apply this theory in studying the growth of entire solutions to difference equations involving the Wilson operator.
We establish a Wiman-Valiron theory of a polynomial series based on the Askey-Wilson operator $\mathcal{D}_q$, where $q\in(0,1)$. For an entire function $f$ of log-order smaller than $2$, this theory includes (i) an estimate which shows that $f$ behaves locally like a polynomial consisting of the terms near the maximal term of its Askey-Wilson series expansion, and (ii) an estimate of $\mathcal{D}_q^n f$ compared to $f$. We then apply this theory in studying the growth of entire solutions to difference equations involving the Askey-Wilson operator.
The paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.
Let $ϕ(x)=\sum α_n x^n$ be a formal power series with real coefficients, and let $D$ denote differentiation. It is shown that "for every real polynomial $f$ there is a positive integer $m_0$ such that $ϕ(D)^mf$ has only real zeros whenever $m\geq m_0$" if and only if "$α_0=0$ or $2α_0α_2 - α_1^2 <0$", and that if $ϕ$ does not represent a Laguerre-Pólya function, then there is a Laguerre-Pólya function $f$ of genus $0$ such that for every positive integer $m$, $ϕ(D)^mf$ represents a real entire function having infnitely many nonreal zeros.
For entire functions $f(z)=\sum_{n=0}^{+\infty}a_nz^n, z\in {\Bbb C},$ P. L${\rm \acute{e}}$vy (1929) established that in the classical Wiman's inequality $M_f(r)\leqμ_f(r)\times $ $\times(\lnμ_f(r))^{1/2+\varepsilon},\ \varepsilon>0,$ which holds outside a set of finite logarithmic measure, the constant 1/2 can be replaced almost surely in some sense, by 1/4; here $M_f(r)=\max\{|f(z)|\colon |z|=r\},\ μ_f(r)=\max\{|a_n|r^n\colon n\geq0\},\ r>0. $ In this paper we prove that the phenomenon discovered by P. L${\rm\acute{e}}$vy holds also in the case of Wiman's inequality for entire functions of …
On the number of real critical points of logarithmic derivatives and the Hawaii conjecture
本文研究了实整函数对数导数的实零点数目关系，通过建立$Q$与$Q_1$实零点数目的联系，成功证明了Hawaii猜想。
We study fluctuations in the number of zeros of random analytic functions given by a Taylor series whose coefficients are independent complex Gaussians. When the functions are entire, we find sharp bounds for the asymptotic growth rate of the variance of the number of zeros in large disks centered at the origin. To obtain a result that holds under no assumptions on the variance of the Taylor coefficients we employ the Wiman-Valiron theory. We demonstrate the sharpness of our bounds by studying well-behaved covariance kernels, which we call admissible (after Hayman).
Rate of growth of random analytic functions, with an application to linear dynamics
本文建立随机整函数与单位圆盘随机解析函数的 Wiman–Valiron 型不等式，改进已有结果，并用于刻画任意混沌加权后移算子频繁超循环函数在例外集外的增长率。
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.
A conjecture of I. Krasikov is proved. Several discrete analogues of classical polynomial inequalities are derived, along with results which allow extensions to a class of transcendental entire functions in the Laguerre-Pólya class.
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.
We resolve a question of Bank and Laine on the zeros of solutions of $w'' + Aw = 0$ where $A$ is an entire function of finite order.
In this paper, we prove a normal criteria for family of meromorphic functions. As an application of that result, we establish a uniqueness theorem for entire function concerning a conjecture of R. Bruck. The above uniqueness theorem is an improvement of a problem studied by L. Z. Yang et. al [14]. However, our method differs the method of L. Z. Yang et. al [14]. We mainly use normal family theory and combine it with Nevanlinna theory instead of using only the Nevanlinna theory as in [14].
Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary
本文在高分歧条件下证明了射影空间中Vojta的abc猜想，并探讨了射影紧致积性变种中的相关结果及Campana轨道猜想的版本。
A theorem of A. and C. Rényi on periodic entire functions states that an entire function $f(z) $ must be periodic if $ P(f(z)) $ is periodic, where $ P(z) $ is a non-constant polynomial. By extending this theorem, we can answer some open questions related to the conjecture of C. C. Yang concerning periodicity of entire functions. Moreover, we give more general forms for this conjecture and we prove, in particular, that $ f(z) $ is periodic if either $ P(f(z)) f^{(k)}(z) $ or $ P(f(z))/f^{(k)}(z) $ is periodic, provided that $ f(z) $ has a finite Picard exceptional value. We also investigate th…
We construct an entire function $f$ with only three singular values whose order of growth can change under a quasiconformal equivalence. This is a counterexample to the Order Conjecture in the Speiser class ${\mathcal S}$ of entire functions.
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…
On the two conjectures
本文针对整函数与其高阶差分算子共享整函数的唯一性问题展开研究，证实了学者提出的两个相关猜想，辅以示例支撑结论。
In the paper, we have shown that every entire solution of the differential difference equation $Δ_η^{m}f-Q_1=(Δ_η^{k}f-Q_{2})e^{P}$ satisfy hyper order of $f=$degree of $P$ and using this result we prove differential difference counter part Brück conjecture. Also we have proved differential difference counter part Brück conjecture for the special case of hyper order of $f<\frac{1}{2}$.
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…
Entire solutions of a certain type differential-difference equation and differential-difference analogue of Bruck conjecture
本文确定了一类微分差分方程有限级整函数解的精确形式，并推广了Brück猜想的微分差分模拟，得到唯一性结果，改进了Liu和Dong的结论。
A conjecture concerning some pairs of interfering estimates for some integrals is formulated in three equivalent versions. Its importance for the the Paley problem for plurisubharmonic functions and for certain classes of extremal problems for entire functions of several variables is declared.
Pursuing McQuillan's philosophy in proving the Green-Griffiths conjecture for certain surfaces of general type, we deal with the algebraic degeneracy of entire curves tangent to holomorphic foliations by curves. Inspired by the recent work of Paun and Sibony, we study the intersection of Ahlfors current with the tangent bundle of the foliation, and derive some consequences. In particular, we introduce the definition of weakly reduced singularities for foliations by curves, which requires less work than the exact classification for foliations. Finally we discuss the strategy to prove the Green-…
Let f be a transcendental entire function that omits a complex value a. We show that for every simply connected region D that does not contain a the full preimage of D is disconnected. We conjecture that the same holds if one only assumes that a is omitted locally. We were able to prove this conjecture under the additional assumption that f is of finite order. We include some auxilliary results on the singularities of the inverses of entire functions which are of independent interest.
Inspired by the work of Bank on the hypertranscendence of $Γe^h$ where $Γ$ is the Euler gamma function and $h$ is an entire function, we investigate when a meromorphic function $fe^g$ cannot satisfy any algebraic differential equation over certain field of meromorphic functions, where $f$ and $g$ are meromorphic and entire on the complex plane, respectively. Our results (Theorem 1 and 2) give partial solutions to Bank's Conjecture (1977) on the hypertranscendence of $Γe^h$. We also give some sufficient conditions for hypertranscendence of meromorphic function of the form $f+g$, $f\cdot g$ and …
A counterexample to Fuchs's conjecture
本文构造了阶为任意$\rho\in(0,1/2)$的整函数，使其对数导数以零为亏值，从而否证了Fuchs的猜想。
The subject of our thesis is the uniqueness theory of meromorphic functions and it is devoted to problems concerning Bruck conjecture, set sharing and related topics. The tool, we used in our discussions is classical Nevanlinna theory of meromorphic functions. In 1996, in order to find the relation between an entire function with its derivative, counterpart sharing one value CM, a famous conjecture was proposed by R. Bruck. Since then the conjecture and its analogous results have been investigated by many researchers and continuous efforts have been put on by them. In our thesis, we have obtai…
On the Conjecture of C. C. Yang and periodicity of meromorphic functions
本文研究杨重骏提出的两个关于整函数周期性的猜想，通过建立新的周期性定理，显著改进并推广了前人的结果到更广泛的亚纯函数类。
Universality for roots of derivatives of entire functions via finite free probability
本文基于有限自由概率论，证明了整个函数导数根在重复求导时趋于均匀分布的猜想，包括对Jensen多项式和Hermite普遍性猜想的贡献。
On Steiner entire function
本文提出并研究了 Steiner 整函数，将经典 Steiner 多项式推广至无限维以编码凸紧集的几何信息，揭示了其解析增长性质对 Gaussian 连续性的刻画，并据此推翻了 Gao 和 Vitale 的猜想。
Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture
本文研究了分离型整个函数的Julia集拓扑性质，证明了其Julia连续体为弧状连续体，并构造了具有特定性质的函数，与Eremenko猜想相关。
Let $f$ and $g$ be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of $f$ and $g$ are equal; in particular, we show that $J(f)=J(g)$ provided that neither $f$ nor $g$ has a simply connected wandering domain in the fast escaping set.
Uniqueness of entire function concerning derivatives and shifts
本文研究整函数关于导数与平移的唯一性问题，提出两个结果，分别解决了Majumder等人提出的公开问题，并在很大程度上改进推广了Huang和Fang的近期结果。
Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite …
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…
In this paper, we study a problem of non-constant entire function $ f $ that shares a set $ \mathcal{S}=\{a,b,c\} $ with its $ k $-th derivative $ f^{(k)} $, where $ a, b $ and $ c $ are any three distinct complex numbers. We have found a gap in the statement of the main result of \textit{Chang-Fang-Zalcman} \cite{Cha & Fan & al-ADM-2007} and with some help of the method used by \textit{Chang-Fang-Zalcman}, we have generalized the result of \textit{Chang-Fang-Zalcman} in a more compact form. As an application, we generalize the famous Br$\ddot{u}$ck conjecture \cite{Bru-1996} with the idea of …
A counterexample to Eremenko's conjecture
本文通过构造逃逸集具有紧致连通分量（包括单点分量）的超越整函数，否证了Eremenko猜想，并面向一般读者介绍该问题及构造变体。
In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, "most" preimages of almost all points are concentrated in a small subset of the plane. In 1988, Lewis and Wu proved Littlewood's conjecture. Using techniques from complex dynamics, we construct entire functions of finite order with a bounded set of singular values for which the set of exceptional preimages is infinite, with logarithmically growing cardinality.
An entire function that violates quasiminimality
针对 Koiran 关于将指数函数替换为任意一元整函数后拟极小性是否仍成立的问题，本文利用全纯逼近理论构造反例，给出否定回答。
We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. In particular, we describe a class of transcendental entire functions for which an analogue of the Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples to show that K(f) can be totally disconnected, and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.
In the work, we focus on a conjecture due to Z.X. Chen and H.X. Yi[1] which is concerning the uniqueness problem of meromorphic functions share three distinct values with their difference operators. We prove that the conjecture is right for meromorphic function of finite order. Meanwhile, a result of J. Zhang and L.W. Liao[10] is generalized from entire functions to meromorphic functions.
In this paper, A. Avila's theorem on convergence of the exact quantization scheme of A. Voros is related to the reality proofs of eigenvalues of certain PT-symmetric boundary value problems. As a result, a special case of a conjecture of C. Bender, S. Boettcher and P. Meisinger on reality of eigenvalues is proved.
Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components.
The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a "small" set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concern…
We show that the fast escaping set $A(f)$ of a transcendental entire function $f$ has a structure known as a spider's web whenever the maximum modulus of $f$ grows below a certain rate. We give examples of entire functions for which the fast escaping set is not a spider's web which show that this growth rate is best possible. By our earlier results, these are the first examples for which the escaping set has a spider's web structure but the fast escaping set does not. These results give new insight into a conjecture of Baker and a conjecture of Eremenko.
We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.
Eremenko's conjecture on higher-order Briot--Bouquet equations
该研究证明了高阶布里奥-布凯方程的埃列缅科猜想，通过多项式复合刚性定理解决了整函数解的分类问题，并得出马勒类函数导数代数无关的算术结论。
In this note, we establish the following Second Main Theorem type estimate for every entire non-algebraically degenerate holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})$, in present of a {\sl generic} hypersuface $D\subset\mathbb{P}^n(\mathbb{C})$ of sufficiently high degree $d\geq 15(5n+1)n^n$: \[ T_f(r) \leq \,N_f^{[1]}(r,D) + O\big(\log T_f(r) + \log r \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,D)$ stand for the order function and the $1$-truncated counting function in Nevanlinna theory. This inequality quantifies recent results on the logarithmic Green--Grif…
Motivated by the recent proof of Newman's conjecture \cite{R-T} we study certain properties of entire caloric functions, namely solutions of the heat equation $\partial_t F = \partial_z^2 F$ which are entire in $z$ and $t$. As a prerequisite, we establish some general properties of the order and type of an entire function. Then, we start our inquiry on entire caloric functions by determining the necessary and sufficient condition for a function $f(z)$ to be the initial condition of an entire solutions of the heat equation and, subsequently, we examine the relation of the $z$-order and $z$-type…
Eremenko's conjecture, wandering Lakes of Wada, and maverick points
本文开发了一种将复平面全闭子集实现为整函数游荡集的通用技术，构造了 Eremenko 猜想的反例，并解决了关于 Fatou 分支共享边界、非逃逸点测度及单连通分支闭包补集等多个复动力学开放问题。
Results on the postsingular set of compositions of transcendental entire functions
研究可交换超越整函数\(f\)与\(g\)（\(g = af^p + b\)）动力学，探讨奇异值和后奇异集在复合下性质，表明若一个函数具有某些性质，另一个及其复合也有，还得出关于超越半群的结果。
Let f be an entire function that has only finitely many critical and asymptotic values. Up to topological equivalence, the function $f$ is determined by combinatorial information, more precisely by an infinite graph known as a "line-complex". In this note, we discuss the natural question whether the order of growth of an entire function is determined by this combinatorial information. The search for conditions that imply a positive answer to this question leads us to the "area property", which turns out to be related to many interesting and important questions in conformal dynamics and functio…
The "pits effect" for entire functions of exponential type and the Wiener spectrum
本文研究了整个函数的零点等分布问题，提出一个简单的谱条件，适用于多种随机和伪随机序列，并揭示了梅比乌斯函数在二元 Chowla 猜想下的性质。
We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide examples of entire functions with an orbit of unbounded fast escaping wandering domains, answering a long-standing question of Rippon and Stallard. Moreover, these examples cover all possible types of simply connected wandering domains in terms of convergence to the boundary. In relation to a conjecture of Baker, it was unknown whether functions of order less than one could have unbounded wandering d…
We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salomão and Vargas. We also prove density of (real) hyperboli…
Sum of the exponential and a polynomial: Singular values and Baker wandering domains
本文研究形如指数函数复合与多项式之和的整函数，证明其无穷远点为Baker遗漏值，具有无限临界值且无有限渐近值，并证明此类函数无Baker游荡域，为相关猜想提供候选研究对象。
For a transcendental entire function $f$, the property that there exists $r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, where $m(r)=\min \{|f(z)|:|z|=r\}$, is related to conjectures of Eremenko and of Baker, for both of which order $1/2$ minimal type is a significant rate of growth. We show that this property holds for functions of order $1/2$ minimal type if the maximum modulus of $f$ has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg's method of constructing entire functions of small growth, which allows…
We confirm Flandrin's prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions), a consequence of the conjectured double honeycomb mean model for the network of zeros and local maxima, where the area of local maxima centered hexagons is three times larger than the area of zero centered hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their expected density of zeros is 1, have an expected density of 5/3 critical points, …
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