Permutable entire functions satisfying algebraic differential equations

具有角密度的整个函数空间的唯一性集，III：如何最小化类型

Uniqueness sets with angular density for spaces of entire functions, III: how to minimize the type

本文研究了整个函数空间中具有角密度的唯一性集，通过分析满足一定正则性条件的离散集Λ，证明存在一个类型最小化的测度Δ₀，其质量少于2ρ。对于ρ=2的情况，确定了临界唯一性类型的几何表达。

The Logarithm of the Modulus of an Entire Function as a Minorant for a Subharmonic Function outside a Small Exceptional Set

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. In 2016, we obtained a result on the existence of an entire function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb C$, where functions ${\sf B}_u$ are integral averages of $u$ for rapidly shrinking disks as it approaches infinity. We give another equivalent version of this result with $\log |f|\leq u$ outside a very small exceptional set if $u$ is of finite order.

具有角密度的整个函数空间的唯一性集，I. 基础

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。

Sampling and interpolation in de Branges spaces with doubling phase

The de Branges spaces of entire functions generalise the classical Paley-Wiener space of square summable bandlimited functions. Specifically, the square norm is computed on the real line with respect to weights given by the values of certain entire functions. For the Paley-Wiener space, this can be chosen to be an exponential function where the phase increases linearly. As our main result, we establish a natural geometric characterisation, in terms of densities, for real sampling and interpolating sequences in the case when the derivative of the phase function merely gives a doubling measure o…

The restriction from below of the subharmonic function by the logarithm of the module of entire function

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is an entire function $f\not\equiv 0$ such that $$ \ln |f(z)|\leq \frac{1}{2π}\int_0^{2π}u(z+r(z)e^{iθ})\,{\mathrm d}θ\quad\text{for all $z\in\mathbb C$.}$$ A similar result is established for subharmonic functions of finite order with inequalities of the form $\ln|f(z)|\leq u(z)$ at all points $z\in\mathbb C\setminus E$, where the exceptional set $E$ is small in terms …

关于函数方程$f^8+g^8+h^8=1$和$f^6+g^6+h^6=1$存在性问题

On existence questions for the functional equations $f^8+g^8+h^8=1$ and $f^6+g^6+h^6=1$

本文证明了对于函数方程$f^8+g^8+h^8=1$和$f^6+g^6+h^6=1$，不存在非常数亚纯或整函数解，回答了G.G.Gundersen的问题。

An entire function connected with the approximation of the golden ratio

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.

On the Heins Theorem

It is known that the famous Heins Theorem (also known as the de Branges Lemma) about the minimum of two entire functions of minimal type does not extend to functions of finite exponential type. We study in detail pairs of entire functions $f, g$ of finite exponential type satisfying $\sup_{z\in\mathbb{C}}\min\{|f(z)|,|g(z)|\}<\infty.$ It turns out that $f$ and $g$ have to be bounded on some rotating half-planes. We also obtain very close upper and lower bounds for possible rotation functions of these half-planes.

Entire functions with prescribed singular values

We introduce a new class of entire functions $\mathcal{E}$ which consists of all $F_0\in\mathcal{O}(\mathbb{C})$ for which there exists a sequence $(F_n)\in \mathcal{O}(\mathbb{C})$ and a sequence $(λ_n)\in\mathbb{C}$ satisfying $F_n(z)=λ_{n+1}e^{F_{n+1}(z)}$ for all $n\geq 0$. This new class is closed under the composition and its is dense in the space of all non-vanishing entire functions. We prove that every closed set $V\subset \mathbb{C}$ containing the origin and at least one more point is the set of singular values of some locally univalent function in $\mathcal{E}$, hence this new clas…

Hardy-type theorem for functions orthogonal with respect to their zeros. The Jacobi weight case

Motivated by G. H. Hardy's 1939 results \cite{Hardy} on functions orthogonal with respect to their real zeros $λ_{n}, n=1,2,... $, we will consider, within the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval $(0,1)$, that is, the functions $f(z)=z^νF(z), ν\in \mathbb{R}$, where $F$ is entire and \begin{equation*} \int_{0}^{1}f(λ_{n}t)f(λ_{m}t)t^α(1-t)^βdt=0,\quad α>-1-2ν, β>-1, \end{equation*}% when $n\neq m$. Considering all possible functions on this class we are lead to the discovery of a new fam…

Erdös 最大模点问题的一个近似解

An approximate solution to Erdös' maximum modulus points problem

针对 Erdös 关于整函数最大模点数是否可趋于无穷的开放问题，本文构造了一个任意接近满足该性质的超越整函数，为肯定答案提供了最强证据。

Interpolation of Power Mappings

Let $(M_j)_{j=1}^\infty\in\mathbb{N}$ and $(r_j)_{j=1}^\infty\in\mathbb{R}^+$ be increasing sequences satisfying some mild rate of growth conditions. We prove that there is an entire function $f: \mathbb{C} \rightarrow\mathbb{C}$ whose behavior in the large annuli $\{ z\in\mathbb{C} : r_{j}\cdot\exp(π/M_{j})\leq|z|\leq r_{j+1}\}$ is given by a perturbed rescaling of $z\mapsto z^{M_j}$, such that the only singular values of $f$ are rescalings of $\pm r_j^{M_j}$. We describe several applications to the dynamics of entire functions.

Order and Hyper-order of Solutions of Second Order Linear Differential Equations

We have discussed the problem of finding the condition on coefficients of $f''+A(z)f'+B(z)f=0, \quad B(z)(\not \equiv 0)$ so that all non-trivial solutions are of infinite order. The hyper-order of these non-trivial solutions of infinite order is also found when $λ(A)<ρ(A)$ and $B(z)$ is a transcendental entire function satisfying some conditions.

关于Xu和Wang的一个公开问题的某些Fermat型PDDEs的解

Solutions for certain Fermat-type PDDEs concerning an open problem of Xu and Wang

本文研究了Xu和Wang提出的关于$\mathbb{C}^2$的公开问题，确定了满足某些Fermat型偏微分-差分方程的有限阶亚纯函数和整函数的存在性与形式，并给出实例验证。

The Complex Monge-Ampère Equation, Zoll Metrics and Algebraization

Let M be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of $TM$. We prove here that the only real analytic Zoll metric on the $n$-sphere with an entire adapted complex structure on $TM$ is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in ter…

A negative minimum modulus theorem and surjectivity of ultradifferential operators

In 1979 I. Ciorănescu and L. Zsidó have proved a minimum modulus theorem for entire functions dominated by the restriction to the positive half axis of a canonical product of genus zero, having all roots on the positive imaginary axis and satisfying a certain condition. Here we prove that the above result is optimal: if a canonical product ω of genus zero, having all roots on the positive imaginary axis, does not satisfy the condition in the 1979 paper, then always there exists an entire function dominated by the restriction to the positive half axis of ω, which does not satisfy the desired mi…

Speiser 与 Misiurewicz 相遇

Speiser meets Misiurewicz

该研究提出超越整函数的Misiurewicz条件，分析满足该条件的Speiser函数的参数空间扰动，证明其可被双曲映射逼近、Julia集零测时为双曲映射的Lebesgue稠密点，且该类函数集参数空间测度为零。

Generalization and development of the Malliavin-Rubel theorem on small entire functions of exponential type with given zeros

Previously, we developed the technique of balayage of measures or charges and ($δ$-)subharmonic functions of finite order onto an closed system of rays $S$ with a vertex at zero on the complex plane $\mathbb C$. In this article, we use only two kinds of balayage of measure and charge, as well as of subharmonic functions of finite type under the order $1$ and their differences. First, it is a classical balayage of the genus $q=0$ on a system of four closed rays: positive and negative, and real and imaginary semi-axis $\mathbb R^+$, $-\mathbb R^+$, $i\mathbb R^+$, $-i\mathbb R$. Second, it is tw…

Real Analytic Bergman Spaces

The usual examples of Bergman spaces consist of the closure of an algebra of holomorphic functions on a domain. One can also take the real part of such functions, but essentially one is looking at the same object. In this paper the author shows that the properties of real analyticity and bounded point evaluation can be preserved under closure in a weighted $L^p$ space, if one takes the algebra generated by $z$ and $(\bar z)g(z)$, where $g$ is an entire function satisfying some condition on the distribution of zeros near $\infty$. The condition is not difficult to satisfy with simple examples. …

On periodicity of a meromorphic function when sharing two sets IM

In this paper, we have investigated the sufficient conditions for periodicity of meromorphic functions and obtained two results directly improving the result of \emph{Bhoosnurmath-Kabbur} \cite{Bho & Kab-2013}, \emph{Qi-Dou-Yang} \cite{Qi & Dou & Yan-ADE-2012} and \emph{Zhang} \cite{Zha-JMMA-2010}. Let $\mathcal{S}_{1}=\left\{z:\displaystyle\int_{0}^{z-a}(t-a)^n(t-b)^4dt+1=0\right\}$ and $\mathcal{S}_{2}=\bigg\{a,b\bigg\}$, where $n\geq 4(n\geq 3)$ be an integer.\emph{Let $f(z)$ be a non-constant meromorphic (entire) function satisfying $\ol E_{f(z)}(\mathcal{S}_j)=\ol E_{f(z+c)}(\mathcal{S}_j…