Kuldeep Singh Charak, Rahul Kumar
We prove a Zalcman-Pang lemma in several complex variables and apply it to obtain several complex variables analogues of the known normality criteria like Lappan's five-point theorem and Schwick's theorem.
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On normality and $φ$-normality of holomorphic functions in several complex variables
本文研究多复变全纯函数关于全微分的φ-正规函数和正规族，给出单位球上全纯函数为φ-正规的充分条件，并利用全微分多项式建立正规性准则，推广了已知结果。
Normality criterion for a family of holomorphic curves that partially share wandering hyperplanes with their derivatives, and holomorphic functions lifted to curves in $P^2(\mathbb{C})$
本研究推广了已有复射影空间全纯曲线族正规性结果，得到亚纯函数族分担游荡函数的正规定则，还通过将全纯函数表示为到P²(C)的映射得到新定则，可构造Bloch原理逆命题的反例。
In this paper, we continue to discuss normality based on a single\linebreak holomorphic function. We obtain the following result. Let $\CF$ be a family of functions holomorphic on a domain $D\subset\mathbb C$. Let $k\ge2$ be an integer and let $h(\not\equiv0)$ be a holomorphic function on $D$, such that $h(z)$ has no common zeros with any $f\in\CF$. Assume also that the following two conditions hold for every $f\in\CF$:\linebreak %{enumerate} [(a)] (a) $f(z)=0\Longrightarrow f'(z)=h(z)$ and %[(b)] (b) $f'(z)=h(z)\Longrightarrow|f^{(k)}(z)|\le c$, where $c$ is a constant. Then $\CF$ is normal o…
Extension of classical Mandelbrojt's criterion for normality of a family of holomorphic zero-free functions of several complex variables is given. We show that a family of holomorphic functions of several complex variables whose corresponding Levi form are uniformly bounded away from zero is normal.
In this paper, we prove some value distribution results which lead to some normality criteria for a family of analytic functions. These results improve some recent results.
In this paper, we prove some normality criteria concerning transitivity of normality from one family of meromorphic functions to another which improve and generalize some recent results. We also prove some value distribution results for certain differential polynomials which lead to some normality criteria involving sharing of holomorphic functions with certain differential polynomials. As a consequence, a counterexample to the converse of the Bloch's principle is also given.
In this paper, we prove normality criteria for families of meromorphic functions involving sharing of a holomorphic function by a certain class of differential polynomials. Results in this paper extends the works of different authors carried out in recent years.
In this paper we prove some normality criteria for a family of meromorphic functions concerning shared analytic functions, which extend or generalized some result obtained by Y. F. Wang, M. L. Fang~\cite{WF} and J. Qui, T. Zhu ~\cite{QZ}.
In this paper we prove some normality criteria for a family of meromorphic functions, which involves the zeros of certain differential polynomials generated by the members of the family.
Some Normality Criteria
本文证明了一族亚纯函数的若干正规性准则，涉及由该族成员生成的某些微分单项式共享一个非零值，推广了Schwick的部分结论。
In this paper, we first generalize a value distribution result of Lahiri and Dewan [4] and as an application of this result we prove a normality criterion using partial sharing of small functions. Further, in sequel normality criteria of Hu and Meng [3] and Ding, Ding and Yuan [1] are improved and generalized when the domain $D:=\{z:|z|<R,0<R\leq\infty\}$.
Normality criteria for a family of meromorphic functions with multiple zeros
本文证明了具有多重零点的亚纯函数族的正规性准则，主要结果涉及微分多项式共享全纯函数，并推广了Fang、Zalcman和Chen等人的已有结果。
In this paper, we prove two normality criteria for a family of meromorphic functions. The first criterion extends a result of Fang and Zalcman[Normal families and shared values of meromorphic functions II, Comput. Methods Funct. Theory, 1(2001), 289 - 299] to a bigger class of differential polynomials whereas the second one leads to some counterexamples to the converse of the Bloch's principle.
We prove two sufficient conditions of quasi-normality in which each pair $f$ and $g$ of $\mathcal{F}$ shares some holomorphic functions.
In this paper we continue our earlier investigations on normal families of meromorphic functions\cite{CS2}. Here, we prove some value distribution results which lead to some normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family and get some recently known results extended and improved. In particular, the main result of this paper leads to a counterexample to the converse of Bloch's principle.
Normal Families of Holomorphic Curves and Sharing of Moving Hyperplanes Wandering on $\mathbb{P}^n$
本文推广了 Schwick 的结果，研究了取值于 $\mathbb{P}^n$ 的全纯曲线族，考虑游动超平面并建立了关于共享超平面的正规性充分条件。
Asymptotic Normality of Divisors of Random Holomorphic Sections on Non-compact Complex Manifolds
本文研究非紧 Hermite 流形上随机全纯截面零因子的统计行为，证明相关光滑线性统计量满足中心极限定理。
Normality and Montel's Theorem
本文提出并证明了一个亚纯函数族的正规性准则，该准则涉及零点重数及族中成员共享全纯函数，并在某种意义上推广了Montel的正规性判别法。
Normality and Sharing Functions
本文证明了涉及全纯函数共享的亚纯函数族正规性准则，推广了H. H. Chen、M. L. Fang、M. Han和Y. Gu的部分结果。
Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $φ(\not\equiv 0,\infty)$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r, f)$ in terms of $N\left(r,\frac{1}{M[f]-φ(z)}\right)$ as well as $\ol{N}\left(r,\frac{1}{M[f]-φ(z)}\right)$, where $M[f]$ is the differential monomial, generated by $f$.\par Moreover, we prove one normality criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq1)$, $q_{0}(\geq 3)$, $q_{i}(\geq0)$ $(i=1,2,\ldots,k-1)$, $q_{k…
In this paper, we obtained some normality criteria for families of holomorphic functions. Which generalizes some results of Fang, Xu, Chen and Hua.
A Criterion for quasi normality in $\mathbb{C}^n$
本文针对取值于完备复Hermitian流形的全纯函数族，提出了一个Zalcman型重正规化结果，以此作为判定其在$\mathbb{C}^n$中拟正规性的准则。
Avoidance Criteria for Normal Holomorphic Curves on Complex Projective Space
研究全纯曲线族在复射影空间中避开足够多移动超曲面的条件，并探讨共享超平面的全纯曲线族的正常性条件。
We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\le C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general counterexample to show that for $α>1$ and $k\ge2$ the condition $$\frac{|f^{(k)}|}{1+|f|^α}(z)\le C \qquad \mbox{ for all } z\in D$$ does not imply quasi-normality.
In this paper, besides a counterexample to Bloch's principle, normality criteria leading to counterexamples to the converse of Bloch's principle in several complex variables are proved. Some Picard-type theorems and their corresponding normality criteria in $\mathbb C^{n}$ are also obtained.
Normality Criteria for Differential Monomials and the Sharpness of Lappan-type Theorems
本文研究微分单项的正常性准则，证明通过更高阶导数可减少目标集点数，并推广了Pang-Zalcman定理以更广泛的应用范围。
A normality criterion for a family of meromorphic functions
本文提出了一类亚纯函数在特定条件下的正常性准则，涉及齐次微分多项式和全纯函数的交互作用。
This paper is motivated by Brolin's theorem. The phenomenon we wish to demonstrate is as follows: if $F$ is a holomorphic correspondence on $\mathbb{P}^1$, then (under certain conditions) $F$ admits a measure $μ_F$ such that, for any point $z$ drawn from a "large" open subset of $\mathbb{P}^1$, $μ_F$ is the weak*-limit of the normalised sums of point masses carried by the pre-images of $z$ under the iterates of $F$. Let ${}^\dagger{F}$ denote the transpose of $F$. Under the condition $d_{top}(F) > d_{top}({}^\dagger{F})$, where $d_{top}$ denotes the topological degree, the above phenomemon was…
A Zalcman-Pang type rescaling result for $ϕ$-normal harmonic mappings and applications
本文研究单位圆盘上调和映射的正规性与$ϕ$-正规性，建立涉及扩展球面导数的充分条件，证明Zalcman-Pang型重标度引理，并导出新的$ϕ$-正规性准则及Lappan型刻画。
By using Nevanlinna theory, we prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials generated by the members of the family.
In the present paper, we introduced the extended bicomplex plane $\bar{\mathbb{T}}$, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about the convergence of the sequences of bicomplex meromorphic functions. Hence the concept of the normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the Fundamental Normality Tests for families of bicomplex holomorphic functions and bico…
Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $ϕ:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a generalization of a classical condition for univalence of Nehari.
Normality Criteria for Quasimeromorphic Mappings: Avoidance and Sharing
本文研究准共形映射的正常性，基于Lappan的正常映射理论，推导出在欧几里得空间中准共形映射的正常家族准则。
On $φ$-Normality of Harmonic Mappings and Their Families
本文为调和映射及其族建立了$\varphi$-正规性的Lappan型准则，涉及高阶球面导数与测试值数量，并推广至多项式情形。
A normality Criterion for a Family of Meromorphic Functions
本文研究了Schwick亚纯函数族正规性准则的相反情形，探讨当导数条件存在零点时函数族的正规性问题。
In this paper, as an application of Zalcman's lemma in $\mathbb{C}^n$, we give a sufficient condition for normality of holomorphic functions of several complex variables, which generalizes previous known one-dimensional criterion of A.J. Lohwater and Ch. Pommerenke \cite[Theorem 1]{MR0338381}.
Extension of Lohwater-Pommerenke's Theorem for strongly-normal Maps
引入全纯曲线和对数调和映射的强正规性概念，推广了Lohwater-Pommerenke关于非强正规映射的重标度刻画，并研究了Bloch映射和小Bloch映射的Zalcman-Pang型重标度结果，进一步扩展到强φ-正规映射。
Normality through sharing of pairs of functions with derivatives
本文改进了Schwick定理，证明在亚纯函数族中，若函数与其导数部分共享三对函数（含去心圆盘内的取值），则族正规，并给出多个更优准则及精确性例子。
We show that a family ${\cal F}$ of meromorphic functions in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f^{(j)}|^α}(z)\ge C \qquad \mbox{for all} z\in D \mbox{and all} f\in {\cal F}$$ (where $k$ and $j$ are integers with $k>j\ge 0$ and $C>0$, $α>1$ are real numbers) is quasi-normal. Furthermore, if all functions in ${\cal F}$ are holomorphic, the order of quasi-normality of ${\cal F}$ is at most $j-1$. The proof relies on the Zalcman rescaling method and previous results on differential inequalities constituting normality.
This paper treats a holomorphic self-mapping f: Omega --> Omega of a bounded domain Omega in a separable Hilbert space H with a fixed point p. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Caratheodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping f and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.
In this paper, we prove two normality criteria for families of some functions concerning shared values, the results generalize those given by Hu and Meng. Some examples are given to show the sharpness of our results.
We give a simple uniqueness criterion (and some derived criteria) for holomorphic Abel functions and show that Kneser's real analytic Abel function of the exponential is subject to this criterion.
On Decompositions of H-holomorphic functions into quaternionic power series
本文基于四元数与复全纯函数的代数相似性，建立四元数幂级数收敛理论，证明H-全纯函数与复全纯函数幂级数展开具有相同收敛半径，并证明H-全纯函数必为H-解析函数。
Meromorphically normal families and a meromorphic Montel-Carathéodory theorem
本文研究了取值于 $\mathbb{P}^n$ 的亚纯映射族的亚纯正规性，受 Fujimoto 工作启发给出了一般充分条件，并据此建立了近期全纯映射正规族相关结果的亚纯类似物及 Montel-Carathéodory 定理的亚纯版本。
For a holomorphic one-form $\mathbfξ$ on a weakly 1-complete manifold $X$ with certain properties, we discussed the connectivity of the pair $(\hat{X}, F^{-1}(z))$, where $π: \hat{X} \to X$ is a covering map and $dF=π^*\mathbfξ$. We also discussed the criteria about when such a manifold $X$ admits a proper holomorphic mapping onto a Riemann surface.
Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc
本文研究了在单位圆盘内全纯自映射的分数迭代交换性扩展的准则，提出了若干充分条件以保证半群元素与映射的交换关系，并分析了映射在半群花瓣中的行为。
Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for holomorphic extension, but we show that meromorphic extension holds on a reduced globally irreducible (not neces…
Harmonic mappings, univalence criteria and a theorem of Lehtinen
研究调和映射的单值性准则，证明调和内半径与单位圆的关系，类比Lehtinen定理提供两个相关准则。
We investigate the relation between holomorphic torus actions on complex manifolds of LCK type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold $(M,J)$ contains a non-real compact torus, then there exists a Vaisman metric on the manifold. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of $M$, then $(M,J)$ admits an LCK metric with positive potential. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structur…
We obtain a generalization of the Burns-Krantz rigidity theorem for holomorphic self-mappings of the unit disk in the spirit of the classical Schwarz-Pick Lemma and its continuous version due to L.Harris via the generation theory for one-parameter semigroups. In particular, we establish geometric and analytic criteria for a holomorphic function on the disk with a boundary null point to be a generator of a semigroup of linear fractional transformations under some relations between three boundary derivatives of the function at this point.
Holomorphic and Formal First Integrals for Foliations of Codimension One on Complex Analytic Space Germs
该研究将Mattei-Moussu经典可积性定理推广到奇异复解析空间，证明二维余维1全纯叶状结构存在全纯首次积分的充要条件，给出非正规例子说明正规性的必要性，结合多种技术推导高维可积性结果。
If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension …
Exploring Topological Transitivity in Families of Functions
本文研究连续与全纯函数族的拓扑传递性，建立多种判定准则，并利用亚纯函数扩张族性质为 Montel 三点定理提供新证明。
It is stated a series of criteria of equicontinuity and normality for classes of space mappings with integral restrictions. It is shown that the found conditions are not only sufficient but also necessary. It is given applications to Sobolev's classes.
Rigidity of entire functions sharing a finite set with their partial derivatives in C^n
研究C^n中整体函数与其偏导数共享三个元素有限集的类，利用正常性准则分析其行为并推导存在条件，扩展到多复变量情形，推广单复变函数结果。
We discuss the Siciak-Zaharjuta extremal function of a real convex body in C^n, a solution of the homogeneous complex Monge-Ampere equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadric curves, and the geometry of such curves allows for the determination of the leaves of…
Let $\mathcal{A}(p)$ be the class consisting of functions $f$ that are holomorphic in $\ID\setminus \{p\}$, $p\in (0,1)$ possessing a simple pole at the point $z=p$ with nonzero residue and normalized by the condition $f(0)=0=f'(0)-1$. In this article, we first prove a sufficient condition for univalency for functions in $\mathcal{A}(p)$. Thereafter, we consider the class denoted by $Σ(p)$ that consists of functions $f \in \mathcal{A}(p)$ that are univalent in $\ID$. We obtain the exact value for $\ds\max_ {f\in Σ(p)}Δ(r,z/f)$, where the Dirichlet integral $Δ(r,z/f)$ is given by $$ Δ(r,z/f)=\d…
Bieberbach conjecture, Bohr radius, Bloch constant and Alexander's theorem in infinite dimensions
该研究考察复 Banach 空间单位球上 $F(x)=f(x)x$ 型全纯映射，建立其几何性质判据，并推广 Bieberbach、Fekete-Szegö、Bloch 与 Alexander 等经典结果。
A particular consequence of the famous Carleson-Hunt theorem is that the Taylor series expansions of bounded holomorphic functions on the open unit disk converge almost everywhere on the boundary, whereas on single points the convergence may fail. In contrast, Bayart, Konyagin, and Queffélec constructed an example of an ordinary Dirichlet series $\sum a_n n^{-s}$, which on the open right half-plane $[Re >0]$ converges pointwise to a bounded, holomorphic function -- but diverges at each point of the imaginary line, although its limit function extends continuously to the closed right half plane.…
Riesz-Kolmogorov type compactness criteria in function spaces with applications
本文提出适用于多种函数空间的Riesz-Kolmogorov型紧性准则，并用于刻画Bergman空间上的紧Toeplitz算子及推导Hardy空间上Hankel算子的紧性。
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