B. N. Khabibullin, F. B. Khabibullin
In this article are improved and refined some of our results prior to our recent article from the journal "Russian Mathematics (Izvestiya VUZ. Matematika)" in 2015 at the expense of our latest results in 2016 on lower estimates of subharmonic functions by the logarithm module nonzero holomorphic function.
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We establish a general uniqueness theorem for subharmonic functions of several variables on a domain. A corollary from this uniqueness theorem for holomorphic functions is formulated in terms of the zero subset of holomorphic functions and restrictions on the growth of functions near the boundary of domain.
We consider inductive limits of weighted spaces of holomorphic functions in the unit ball of $\mathbb C^n$. The relationship between sets of uniqueness, weakly sufficient sets and sampling sets in these spaces is studied. In particular, the equivalence of these sets, under general conditions of the weights, is obtained.
In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersects $O(k^3)$ many totally geodesic hypersurfaces.
On Mok's uniqueness problem of holomorphic isometries from complex unit balls of maximal dimension into classical domains
该研究解决了Mok提出的问题，证明了从极大维复单位球到特定经典有界对称域的全纯等距嵌入均等价于Mok嵌入。
In this paper, we prove some fundamental theorems for holomorphic curves on angular domain intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex projective variety with the level of truncation. As applications of the second main theorems for an angle, we will discuss the uniqueness problem of holomorphic curves in an angle instead of the whole complex plane. Detail, we establish a result for uniqueness problem of holomorphic curve by inverse image of a hypersurface. In my knowledge, this is the firs…
Boundary Values and Boundary Uniqueness of $J$-Holomorphic Mapping
本文对在适当意义下有界的$J$-全纯映射建立了一个Fatou型定理，并证明了其零点集的Blaschke条件.证明了$J$-全纯映射对的一个Privalov型唯一性定理。
In this paper, we prove Huang et al.'s conjecture stated that if $f$ is a holomorphic function on $Δ^+:=\{z\in \mathbb C \colon |z|<1,~\mathrm{Im}(z)>0\}$ with $\mathcal{C}^\infty$-smooth extension up to $(-1,1)$ such that $f$ maps $(-1,1)$ into a cone $Γ_C:=\{z\in \mathbb C\colon |\mathrm{Im} (z)| \leq C|\mathrm{Re} (z)|\}$, for some positive number $C$, and $f$ vanishes to infinite order at $0$, then $f$ vanishes identically. In addition, some regularity properties of the Riemann mapping functions on the boundary and an example concerning Huang et al.'s conjecture are also given.
Uniqueness of Holomorphic Abel Functions at a Complex Fixed Point Pair
本文给出了全纯Abel函数的一个简单的唯一性判据（以及一些导出的判据），并证明了指数型的Kneser实解析Abel函数也服从这个判据。
On removable sets for holomorphic functions
本文对紧平面集关于各类全纯函数的可去性进行了全面的研究。我们还讨论了一些应用和几个开放的问题，其中一些是新的。
Isolated fixed point sets for holomorphic maps
研究了复流形上全纯自映射的离散不动点集。主要的注意力集中在这个集合的基数和它的配置。作为一个结果，我们的观察，${\Bbb C}^n$中的有界域没有非平凡的全纯收缩构造。
On behaviour of holomorphically contractible systems under non-monotonic sequences of sets
给出了全纯可缩系统作为集函数关于非单调集合序列的连续性的新结果。特别地，给出了Kobayashi和Carathéodory伪距离以及Lempert和格林函数关于收敛于Hausdorff度量的域序列的连续性.
This paper studies the determining sets for analytic functions from the symmetrized bidisk into the open unit disk in $\mathbb C$. It relates the idea to the uniqueness of the solutions of a Nevanlinna-Pick interpolation problem. It also investigates when certain thin sets can be determining.
In this paper, we give some extension of fundamental theorems in Nevanlinna - Cartan theory for holomorphic curve on M punctured complex planes. As an application, we establish a result for uniqueness problem of holomorphic curve by inverse image of a hypersurface, it is improvement of some results before [8, 14] in this trend.
Approximation of sets defined by polynomials with holomorphic coefficients
设X是一个由多项式定义的解析集，其系数a_1，.，a_s是全纯函数。我们用公式表示条件，使得对于所有序列{a_（1，n）}，.，{a_（s，n）}局部一致收敛于a_1，. A_s分别为真。如果a_（1，n），.，a_（s，n）满足条件，则多项式中a_（j，n）代替a_j得到的集合序列{X_n}收敛于X。
We prove that if f is a holomorphic function on the open unit disc in C whose cluster set C(f) has finite linear measure and is such that the complement of C(f) has finitely many components, then the derivative of f belongs to the Hardy space H^1.
Critical sets of proper holomorphic mappings
证明了：若一个真全纯映射$f: \mathbb C^n \to \mathbb C^N$，$1<n\le N$，将一个有限型伪凸实解析超曲面送入另一个这样的超曲面，则$f$的临界轨迹的任何$n-1$维分量与$M$的两边相交.我们将这一结果应用于$\mathbb C^n$中有界区域间真全纯映射的边界正则性问题。
In our previous work, for a given Riemann surface $Y_0$ with marked handle, we investigated geometric properties of the set of marked once-holed tori $X$ allowing holomorphic mappings of $X$ into $Y_0$. It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.
Let $m,n\geq 1$ are integers and $D$ be a domain in the $$ $\mathbb C^n$ or in the $m$-dimensional real space $\mathbb R^m$. We build positive subharmonic functions on $D$ vanishing on the boundary $\partial D$ of $D$. We use such (test) functions to study the distribution of zero sets of holomorphic functions $f$ on $D\subset \mathbb C^n$ with restrictions on the growth of $f$ near the boundary $\partial D$.
Fixed points and Determining Sets for Holomorphic Self-Maps of a Hyperbolic Manifold
研究了复流形上全纯自同构（和自同态）的不动点集。我们感兴趣的主要目的是确定迫使自同构（自同态）成为恒等式的不动点的数目和配置。这些问题已在一些文件中审查的有界域${\Bbb C}^n$。在这里，我们解决了一般有限维双曲流形的情况。我们还证明了对于非双曲流形的结果是显著不同的。
We consider a problem of whether a property of holomorphic curves on a subset $X$ of the complex plane can be extended to the whole complex plane. In this paper, the property we consider is uniqueness of holomorphic curves. We introduce the propagation set. Simply speaking, $X$ is a propagation set if linear relation of holomorphic curves on the part of preimage of hyperplanes contained in $X$ can be extended to the whole complex plane. If the holomorphic curves are of infinite order, we prove the existence of a propagation set which is the union of a sequence of disks (In fact, the method app…
Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for any path p:[0,1)-->U such that |p(t)|-->1 as t-->1, the path t--> f(p(t)) from [0,1) to B has infinite length.
On the holomorphic closure dimension of real analytic sets
给出一个真实的分析（或者，更一般地，半解析）集合R在n维复空间中，对于R的闭包中的每个点p，唯一的最小复解析芽X_p，它包含芽R_p，我们称X_p的复维数为p是R在p处的全纯闭包维数。我们证明了一个不可约R的全纯闭包维数在一个闭R的补上是常数R的真解析子集，并讨论了它的维数与R的CR维数之间的关系。
Stein neighborhoods of graphs of holomorphic mappings
本文给出了复流形中紧集上的全纯映射的图具有Stein邻域的充分条件。我们证明了在这些条件下，映射具有类似于$\mathbb C^n$中紧集上全纯函数的性质。
Let $D$ be a nonempty domain in $\mathbb C^n$. We give a scale of necessary conditions for the distribution of the zero set of holomorphic function $f$ on domain $D\subset {\mathbb C}^n$ under a restriction on its growth $|f|\leq \exp M$, where $M\not\equiv -\infty$ is a subharmonic function. If $n=1$, $D\neq \mathbb C$ is simply connected, and $M$ is continuous, then this conditions are sufficient.
Uniqueness theorems are considered for various types of almost periodic objects: functions, measures, distributions, multisets, holomorphic and meromorphic functions.
Holomorphic motion for Julia sets of holomorphic families of endomorphisms of CP(k)
在各种等价的稳定性概念下，我们建立了CP（k）的自同态的全纯族的朱利亚集的可测全纯运动.
Proper Holomorphic Embeddings of complements of large Cantor sets in $\Bbb C^2$
本研究针对$\Bbb C^2$中大康托尔集的补集，构建了真全纯嵌入结构，可实现勒贝格测度任意接近4的康托尔集补集的全纯嵌入，拓展了复空间中康托尔集补集的全纯嵌入构造方法。
Let $D$ be a domain in the complex plane $\mathbb C$. It follows from first part of our work that if a non-zero holomorphic function $f$ on $D$ vanishes on a sequence ${\sf Z}\subset D$ and satisfies $|f|\leq M$ on $D$, where $M$ is a subharmonic function on $D$, then the the distribution of ${\sf Z}$ is subordinated to the Riesz measure $ν_M$ of $M$ in a certain sense. Here we show that this result is "almost reversible".
Uniqueness of meromorphic functions that share Two Sets
本文引入新型有限值域集合对，研究亚纯函数由两个有限分担集合唯一确定的问题。
Extended explanation of Orevkov's paper on proper holomorphic embeddings of complements of Cantor sets in $\Bbb C^2$ and a discussion of their measure
本研究阐释了Orevkov论文中$\bbb P^1$去掉康托集后到$\bbb C^2$的逆紧全纯嵌入构造细节，证明该构造可实现康托集豪斯多夫维数为零。
Let $(ϕ_t)_{t \geq 0}$ be a semigroup of holomorphic functions in the unit disk $\mathbb{D}$ and $K$ a compact subset of $\mathbb{D}$. We investigate the conditions under which the backward orbit of $K$ under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of $K$ are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that $K$ forms with the unit disk.
We characterize zero sets for which every subset remains a zero set too in the Fock space $\mathcal{F}^p$, $1\leq p<\infty$. We are also interested in the study of a stability problem for some examples of uniqueness set with zero excess in Fock spaces.
Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds II
我们在Stein流形覆盖上的某些全纯函数代数中建立了复变函数论的基本结果（例如管域上的波尔全纯殆周期函数代数或所有纤维有界全纯函数的代数，例如，在$H^\infty$的电晕问题中）。特别是，在这种情况下，我们得到的结果全纯扩张从复杂的子流形，性质的因子，冠型定理，全纯类似的Peter-Weyl逼近定理，Hartogs型定理，特征的唯一性集等，我们的证明是基于类似的Cartan定理A和B的相干型层的极大理想空间，这些代数证明在第一部分。
In this paper, we prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share three sets. The obtained results improve some recent existing results.
Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds I
我们在Stein流形覆盖上的某些全纯函数代数中发展了复变函数理论。这，特别是，包括结果全纯扩展从复杂的子流形，电晕型定理，性质的因子，全纯类似物的彼得-Weyl逼近定理，Hartogs型定理，表征的唯一性集。这些代数的模型例子是：（1）管域上全纯殆周期函数的玻尔代数;（2）所有纤维有界全纯函数的代数（例如，在$H^\infty$的电晕问题中出现）。我们的方法是基于经典的Oka-Cartan理论的扩展到这些代数的极大理想空间上的凝聚型层--具有复流形的某些特征的拓扑空间。
In this paper we have mainly focused on the uniqueness of two meromorphic functions under the realm of two shared sets problem with certain restrictions which in turn improve the results of { P. Sahoo and H. Karmakar, Uniqueness theorems related to weighted sharing of two sets, Acta Univ. Sapientiae, Mathematica, 10(2) (2018), 329-339} and {P. Sahoo and A. Sarkar, On the uniqueness of meromorphic functions sharing two sets, Bol. Soc. Mat. Mex., (2019), 1-7}. In particular, pointing out some gaps in the statements of the results in { P. Sahoo and H. Karmakar, Uniqueness theorems related to weig…
Uniqueness in Rough Almost Complex Structures and Differential Inequalities
证明了对于Hölder类至少为1/2的殆复结构，在某个非空开集上为常数的J-全纯圆是常数.这与众所周知的，平凡的，非唯一性的结果形成鲜明对比。我们还调查了唯一性问题（做一些开集消失，或消失到无限阶，或有一个非孤立的零，意味着消失）与微分不等式中出现的理论殆复流形。向量值函数的情况不同于标量值函数的情况。
We consider convergence sets of formal power series of the form $f(z,t)=\sum_{n=0}^{\infty} f_n(z)t^n$, where $f_n(z)$ are holomorphic functions on a domain $Ω$ in $\mathbb{C}$. A subset $E$ of $Ω$ is said to be a convergence set in $Ω$ if there is a series $f(z,t)$ such that $E$ is exactly the set of points $z$ for which $f(z,t)$ converges as a power series in a single variable $t$ in some neighborhood of the origin. A $σ$-convex set is defined to be the union of a countable collection of polynomially convex compact subsets. We prove that a subset of $\mathbb{C}$ is a convergence set if and o…
Algebraicity of holomorphic mappings between real algebraic sets in ${\bold C}^n$
本文给出了$\Bbb C^N$中的全纯映射的芽是代数的条件。设$A\subset \bC^N$是一个不可约的实代数集.设存在$\po \in A$使得$A$是$\po$上的极小、一般、全纯非退化子流形.证明了如果$H$是从$\bC^N$到自身的全纯映射的在$p_1 \in A$的芽，且Jacobian $H$不等同于$0$，且$H(A)$包含在与$A$相同维数的实代数集中，则$H$必须作为代数映射扩张到所有$\bC^N$（减去一个复代数集）.相反，我们表明，对于任何"模型情况“（即，$A$由拟齐次实多项式给出），则$A$上的条件实际上是结论成立所必需的。
Let $D$ be a proper domain in the extended complex plane ${\mathbb C}_{\infty}:={\mathbb C}\cup \{\infty\}$, $M=M_+-M_-\not\equiv \pm \infty$ be a difference of non-trivial subharmonic functions $M_{\pm}\not\equiv \mp \infty$ on $D$, $\text{Hol}(D,M)$ be the class of holomorphic function $f$ on $D$ satisfying $|f|\leq \text{const}_f\exp M$ om $D$, ${\sf Z}\subset D$ be a sequence of points in $D$ without limits points in $D$. We give a complete description of the conditions under which the sequence $\sf Z$ is a sequence of all zeros for some nonzero function $f\in \text{Hol}(D,M)$.
In this paper we prove a number of results concerning uniqueness of a meromorphic function as well as its derivative sharing one or two sets. In particular, we deal with the specific question raised in [18], [19], [20] and ultimately improve the result of Banerjee-Bhattacharjee [4].
Uniqueness sets with angular density for spaces of entire functions, II: $ρ=1$
本文研究指数型整函数空间中具有角密度的唯一性集合，在放弃Lindelöf条件限制的情况下展开分析，并保持内容自足可独立阅读。
In this paper, we investigate the value distribution for linear q-difference polynomials of transcendental meromorphic functions of zero order which improves the results of Xu, Liu and Cao (\cite{Xu & Liu & Cao & 2015}). We also investigate the uniqueness of zero order meromorphic function with its q-difference operator sharing two sets with finite weight. Some examples have been exhibited which are relevant to the content of the paper.
Algebraic approximation of analytic sets and mappings
设{X_n}是在全纯链意义下收敛于某个解析集X的解析集序列。我们引入了一个条件，这个条件意味着X的每一个不可约分支都是{X_n}集合的不可约分支序列的极限。其次，我们应用该条件，用Nash解来逼近Nash方程组Q（x，y）=0的全纯解y=f（x）.所提出的方法允许构造一个算法的全纯解的近似。
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。
Let $M$ be a subharmonic function with Riesz measure $ν_M$ in a domain $D$ in the $n$-dimensional complex Euclidean space $\mathbb C^n$, and let $f$ be a nonzero function that is holomorphic in $D$, vanishes on a set ${\sf Z}\subset D$, and satisfies $|f| \leq \exp M$ on $D$. Then restrictions on the growth of $ν_M$ near the boundary of $D$ imply certain restrictions on the area/volume of $\sf Z$. We give a quantitative study of this phenomenon in the subharmonic framework.
In this paper, on the basis of a specific question raised in [6], we further continue our investigations on the uniqueness of a meromorphic function with its higher derivatives sharing two sets and answer the question affirmatively. Moreover, we exhibit some examples to show the sharpness of some conditions used in our main result.
In this paper, we investigate the uniqueness property of meromorphic functions together with its linear difference polynomial sharing two sets. Using the polynomial introduced in [FILOMAT 33(18)(2019), 6055-6072], we have improved the result of Li-Chen [Abstract and Applied Analysis, 2014, Article ID 894968] in sense of reducing cardinalities of the main set S and the associated weights. Some examples have been exhibited to validate our certain claims in the main result.
Ensembles d'unicite pour les polynomes (Sets of Uniqueness for polynomials)
设E是C的正对数容量紧集.让我们假设对于每个多项式$P\not=id$，我们有$P^{-1}(E)\not=E$。则对所有无常数多项式f和g使得$f^{-1}(E)=g^{-1}(E)$，我们有f=g。
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。
Zero (sub)sets for spaces of holomorphic functions and (sub)harmonic minorants
利用平衡函数和格林函数，得到了复平面上区域内全纯函数的加权空间的点列为零点集的一般条件.
Runge approximation on convex sets implies the Oka property
证明了复流形Y上关于Stein流形到Y的全纯映射的存在性和同伦分类的经典Oka性质等价于关于欧几里得空间中紧凸集到Y的全纯映射的Runge逼近性质.
Dual Approach to Certain Questions for Weighted Spaces of Holomorphic Functions
本文在复平面上的区域上的一元全纯函数的加权空间中提出了新的对偶问题，即：给定空间的非平凡性，零点集的描述，（非）唯一性集的描述，亚纯函数作为给定空间的全纯函数的商的表示.
Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint $|f|\leq \exp M$ on $\mathbb B$, where $M\not\equiv \pm \infty$ is $δ$-subharmonic function on $\mathbb B$ with Riesz charge $μ_M$. We give a scale of integral uniform constraints from above on the distribution of the set ${\sf Z}$ via the charge $ν_M$ in terms of $(2n-2)$-Hausdorff measure of the set $\sf Z$, as well as test convex radial functions and $ρ$-subsp…
Proper holomorphic embeddings with small limit sets
该研究针对Stein流形的逆紧全纯嵌入问题，构造出满足增长控制条件的嵌入，且其无穷远极限集可为复射影空间中的线性嵌入副本，推进了多复变全纯嵌入理论。
Envelopes of open sets and extending holomorphic functions on dual Banach spaces
研究了与扩张全纯函数有关的对偶巴拿赫空间中开集的包络。我们给出了各种绝对凸集的例子，表明在许多情况下，扩展是不可能的。我们还建立连接的研究迭代弱 * 序列闭包的凸集的对偶可分空间。
With the help of the notion of weighted sharing of sets, this paper dealt with the question posed by \emph{Yi} \cite{Yi-SC-1994} regarding the uniqueness of meromorphic functions concerning three set sharing. A result has been proved which significantly improved the recent results of {Banerjee - Ahamed} [Bull. Polish Acad. Sci., {62}(3)(2014), 243--256.], \{Banerjee - Mukherjee} [Hokkaido Math. J., {37}(2008), 507--530.] and {Banerjee - Majumder} [Analysis, {34}(2)(2014), 143--162.] by relaxing the nature of sharing. Several examples have been exhibited to show the sharpness of the cardinaliti…
In this article, we explain the link between Pohlen's extended Hadamard product and the holomorphic cohomological convolution on $\mathbb{C}^*$. For this purpose, we introduce a generalized Hadamard product, which is defined even if the holomorphic functions do not vanish at infinity, as well as a notion of strongly convolvable sets.
Entire curves avoiding given sets in C^n
设$F\subset\Bbb C^n$是$\Bbb C^n$的真闭子集，且$A\subset\Bbb C^n\setminus F$至多可数（$n\geq 2$）.给出了$F$和$A$的条件，在此条件下，存在一个全纯浸入（或一个真全纯嵌入）$ϕ:\Bbb C\to\Bbb C^n$与$A\subsetϕ(\Bbb C)\subset\Bbb C^n\setminus F$ .
We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a question raised by Hakobyan and Herron. The metrically removable sets are shown to be related to other classes of "thin" sets that appeared in the literature. They are also related to the removability problems for classes of holomorphic functions with restrictions on the derivative.
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