Gerd Dethloff, Tran Van Tan
In this paper, we give some results on the number of meromorphic mappings of C^m into P^n under a condition on the inverse images of hyperplanes in P^n. At the same time, we give an answer for an open question by H.Fujimoto.
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In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of C^m into P^n with (3n+1) moving targets and truncated multiplicities.
In this paper, we establish a uniqueness theorem for algebraically nondegenerate meromorphic maps of C^m into C P^n and slowly moving hypersurfaces Q_j in C P^n, j=1,...,q in (weakly) general position, where q depends effectively on n and on the degrees d_j of the hypersurfaces Q_j.
In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of C^m into P^n with truncated multiplicities and "few" targets. We also give a theorem of linear degeneration for such maps with truncated multiplicities and moving targets.
The purpose of this article is to show uniqueness theorems for meromorphic mappings of C^m to CP^n with few hyperplanes H_j, j=1,...,q. It is well known that uniqueness theorems hold for q \geq 3n+2. In this paper we show that for every nonnegative integer c there exists a positive integer N(c), depending only on c in an explicit way, such that uniqueness theorems hold if q\geq (3n+2 -c) and n\geq N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n>>0. At the same time, a big number of recent uniqu…
In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate.
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。
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.
Some results in the uniqueness of meromorphic function
该研究札记针对亚纯函数唯一性中的遗留问题，借助 Yamanoi 的深刻结果展开分析并给出若干新结论。
The purpose of this article is to study the uniqueness problem for meromorphic mappings from $\mathbb{C}^{n}$ into the complex projective space $\mathbb{P}^{N}(\mathbb{C}).$ By making using of the method of dealing with multiple values due to L. Yang and the technique of Dethloff-Quang-Tan respectively, we obtain two general uniqueness theorems which improve and extend some known results of meromorphic mappings sharing hyperplanes in general position.
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.
The purpose of this article is to deal with the multiple values and uniqueness problem of meromorphic mappings from $\mathbb{C}^{m}$ into the complex projective space $\mathbb{P}^{n}(\mathbb{C})$ sharing fixed and moving hypersurfaces. We obtain several uniqueness theorems which improve and extend some known results.
Further Investigations on Weighted Value Sharing and Uniqueness of Meromorphic Functions
本文探讨了两个非常数亚纯函数以指定权重二分担一个值时的不同结果，深化了对加权值分担与亚纯函数唯一性问题的研究。
This paper is devoted to the uniqueness problem of the power of a meromorphic function with its differential polynomial sharing a set. Our result will extend a number of results obtained in the theory of normal families. Some questions are posed for future research.
The purpose of this paper has twofold. The first is to prove a unicity theorem for meromorphic mappings of a complete Kähler manifold M in P^n(C) sharing few hypersurfaces. The second is to give a unicity theorem for the case of differential nondegenerate meromorphic mappings sharing 2N-n+3+2ρ(2N-n+1) hyperplanes in N subgeneral position.
In this paper, we continue to study the sharing value problems for higher order derivatives of meromorphic functions with its linear difference and $q$-difference operators. Some of our results generalize and improve the results of Meng--Liu (J. Appl. Math. and Informatics, 37(2019), 133--148) to a large extent.
In this article, by introducing a new method in estimating the counting function of the auxiliary function, we prove a new generalization of uniqueness theorems for meromorphic mappings into $¶^n(\C )$ which share few hyperplanes regardless of multiplicities. Our result improves the previous result in this topic. Moreover our proof is simpler than the previous proofs.
In the existing literature, many researchers consider the uniqueness of the power of a meromorphic function with its derivative counterpart share certain values or small functions. Here we consider the same problem under the aegis of a more general settings namely set sharing
In this paper, we will consider normality and uniqueness property of a family $\mathcal{F}$ of meromorphic functions when $[Q(f)]^{(k)}$ and $[Q(g)]^{(k)}$ share $α$ ignoring multiplicities, for any $f,g\in \mathcal{F}$, where $Q$ is a polynomial and $α$ is a small function. Our results do not need all of zeros of $Q$ have large order as other authors's results.
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].
This article deals with the multiple values and algebraic dependences problem of meromorphic mappings sharing moving hyperplanes in projective space. We give some algebraic dependences theorems for meromorphic mappings sharing moving hyperplanes without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Basing on these results, some unicity theorems regardless of multiplicity for meromorphic mappings in several complex variables are given. These results are extensions and strong improvements of some recent results.
We prove in this note a result on extension of meromorphic mappings, which can be considered as a direct generalisation of the Hartogs extension theorem for holomorphic functions. Namely: THEOREM. Every meromorphic mapping $f:H_n^q(r)\to Y$, where $Y$ is a $q$ - -complete complex space, extends to a meromorphic mapping from $Δ^{n+q}$ to $Y$. Here $H_n^q(r):=Δ^n\times (Δ^q\setminus \barΔ_r^q)\cup Δ_r^n\times Δ^q$ is a "q-concave" Hartogs figure in $C^{n+q}$. Remark that in the case $q=1$, i.e. when $Y$ is Stein, the statement of the Theorem is exactly the Theorem of Hartogs.
In this article, we establish some new second main theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and moving hypersurfaces with truncated counting functions. A uniqueness theorem for these mappings sharing few moving hypersurfaces without counting multiplicity is also given. This result is an improvement of the recent result of Dethloff - Tan [3]. Moreover the meromorphic mappings in our result may be algebraically degenerate. The last purpose of this article is to study uniqueness problem in the case where the meromorphic mappings agree on small identical set…
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 study the uniqueness of the differential-difference of meromorphic functions. We prove the following result: Let $f$ be a nonconstant meromorphic function of $ρ_{2}(f)<1$, let $η$ be a non-zero complex number, $n\geq1, k\geq0$ two integers and let $a\not\equiv0,\infty$ be a small function of $f$. If $f$ and $(Δ_η^{n}f)^{(k)}$ share $0,\infty$ CM and share $a$ IM, then $f\equiv(Δ_η^{n}f)^{(k)}$, which use a completely different method to improve some results due to Chen-Xu [1].
In this paper, we prove some uniqueness results which improve and generalize several earlier works. Also, we prove a value distribution result concerning $f^{(k)}$ which provides a partial answer to a question of Fang and Wang [A note on the conjectures of Hayman, Mues and Gol'dberg, Comp. Methods, Funct. Theory (2013)13, 533-543].
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.
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…
We prove that a meromorphic map defined on the complement of a compact subset of a three-dimensional Stein manifold M and with values in a compact complex three-fold X extends to the complement of a finite set of points. If X is simply connected, then the map extends to all of M.
In this paper, we have investigated the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing a small function. Our results radically extended and improved the results of Bhoosnurmath-Pujari and Harina - Anand not only by sharing small function instead of fixed point but also reducing the lower bound of $ n $. The authors Harina-Anand made plenty of mistakes in their paper. We have corrected all of them in a more convenient way. At last some open questions have been posed for further study in this direction.
Uniqueness theorems for meromorphic inner functions and canonical systems
本文研究上半平面亚纯内函数的唯一性，利用谱、导数值与Clark测度等数据，推广Borg-Levinson两谱定理并推动典范Hamilton系统逆谱理论。
In this paper, taking the question of Zhang and Lü into the background, we present one theorem which will improve and extend some recent results related to the Brück Conjecture.
A General Characterization on the Uniqueness Problem of L-Functions and General Meromorphic Functions
本文提出L-函数和一般整函数唯一性问题的通用标准，扩展了现有结果并提供了广泛应用的证明。
We construct a (non Kähler) compact complex 3-dimensional manifold $X$ having two following properties: 1) for any domain $D$ in $C^2$ every meromorphic map $f$ from this domain into $X$ extends to a meromorphic map from the envelope of meromorphy $\hat D$ of $D$ into $X$; 2) but there exist a meromorphic map $F$ from a punctured ball $B_*$ in $C^3$ into $X$ which doens't extend meromorphically to the origin. In other words, one can allways remove the singularities of complex codimesion two for the meromorphic maps into this $X$, but only up to some subset of complex codimension three. A descr…
Uniqueness of meromorphic function sharing three small functions CM with its $n-$ exact difference
本文证明了超阶小于1的亚纯函数若与其n阶精确差分分担三个小函数（其中两个为η周期）CM，则二者恒等，从而给出差分唯一性定理。
The purpose of this paper is to obtain some sufficient conditions to determine the relation between a meromorphic function and an L-function when certain differential polynomial generated by them sharing a one degree polynomial. The main theorem of the paper extends and improves all the results in {W. J. Hao and J. F. Chen, Uniqueness of L-functions concerning certain differential polynomials, Discrete Dyn. Nat. Soc., 2018, DOI. 10.1155/2018/4673165}, {F. Liu, X. M. Li and H. X. Yi, Value distribution of L-functions concerning shared values and certain differential polynomials, Proc. Japan. Ac…
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…
A new degenerated second main theorem for meromorphic mappings with hypersurfaces
本文为代数非退化亚纯映射与次一般位置超曲面族建立了带截断计数函数的第二基本定理，得到更优的总亏量上界，截断级别独立于超曲面数量，且仅需弱贝祖性质。
The purpose of the paper is to rectify a series of errors occurred in [2], [17], [20] for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing, which is stronger than the usual one. We manipulate the newly introduced notion in this specific section of literature to resolve all the complications. Not only that we have subtly used the same sharing form to a well known unique range set [3] to settle a long time unsolved problem.
We prove that a family of meromorphic mappings from a bidisc to a compact complex surface, which are equicontinuous in a neighborhood of the boundary of the bidisc, has the volumes of its graphs locally uniformly bounded.
Second main theorem and uniqueness problem of meromorphic functions with finite growth index sharing five small functions on a complex disc
本文为复圆盘上具有有限增长指数的亚纯函数建立了截断计数至1层的第二主定理，并推广改进了Nevanlinna五值定理至五个小函数的情形。
The purpose of the paper is to study the uniqueness problem of a $L$ function in the Selberg class sharing one or two sets with an arbitrary meromorphic function having finite poles. We manipulate the notion of weighted sharing of sets to improve one result of Yuan-Li-Yi [Value distribution of $L$-functions and uniqueness questions of F. Gross, Lithuanian Math. J., $\mathbf{58(2)}$(2018), 249-262]. More importantly, we have pointed out a number of gaps in all the results of Sahoo-Halder [Results on $L$ functions and certain uniqueness question of Gross, Lithuanian Math. J., $\mathbf{60(1)}$(20…
Starting with a question of Yuan-Li-Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lithuanian Math. J., 58(2)(2018), 249-262] we have studied the uniqueness of a meromorphic function f and an L-function L sharing two finite sets. At the time of execution of our work, we have pointed out a serious lacuna in the proof of a recent result of a of Sahoo-Halder [ Some results on L-functions related to sharing two finite sets, Comput. Methods Funct. Theo., 19(2019), 601-612] which makes most of the part of the Sahoo-Halder's paper under question. In context of our choice …
Replaced by a shorter version which appeared in press.
In this article we prove a sufficient condition of quasi-normality in higher dimension for a family of meromorphic mappings in which each pair of functions of family shares some moving hypersurfaces. We also prove a normality criterion concerning shared moving hypersurfaces.
If E is a nonempty closed subset of the locally finite Hausdorff (2n-2)-measure on an n-dimensional complex manifold M and all points of E are nonremovable for a meromorphic mapping of M \ E into a compact Kähler manifold, then E is a pure (n-1)-dimensional complex analytic subset of M.
We give several unequivalent notions of convergency of meromorphic functions and more generally meromorphic mappings (strong, weak, $Γ$-convergency and some others). Relations between them are investigated. A version of Rouche theorem for the strongly converging sequences of meromorphic mappings is given. Further we investigate the sets of normality of families of meromorphic mappings, connections with extension properties of those mappings, some apriori estimates for the volume. Finally we apply the technique developped to the study of Fatou sets of meromorphic endomorphisms of Kähler surface…
In this article, we show some uniqueness theorems for meromorphic mappings of $\C^n$ into the complex projective space $\pnc$ sharing different families of moving hyperplanes regardless of multiplicites, where all intersecting points between these mappings and moving hyperplanes with multiplicities more than a certain number do not need to be counted.
The purpose of this paper is to prove the finiteness theorems for meromorphic mappings of a complete connected Kähler manifold into projective space sharing few hyperplanes in subgeneral position without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Our results are extensions and generalizations of some recent ones.
Uniqueness of first derivatives and differences in meromorphic functions and the characterization of entire function periodicity
本文研究亚纯函数一阶导数与差分分担值的唯一性，解决Qi等人开放问题，并给出超越整函数周期性的充分条件。
In this article, we show some new second main theorems for the mappings and moving hyperplanes of $¶^n(\C)$ with truncated counting functions. Our results are improvements of recent previous second main theorems for moving hyperplanes with the truncated (to level $n$) counting functions. As their application, we prove a unicity theorem for meromorphic mappings sharing moving hyperplanes.
We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in $\cc^2$ to a compact subset of $\cc^N$ which doesn't contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions.
In this paper, we prove a general second main theorem for meromorphic mappings into a subvariety $V$ of $\mathbb P^N(\mathbb C)$ with an arbitrary family of moving hypersurfaces. Our second main theorem generalizes and improves all previous results for meromorphic mappings with moving hypersurfaces, in particular for meromorphic mappings and families of moving hypersurfaces in subgeneral position. The method of our proof is different from that of previous authors used for the case of moving hypersurfaces.
In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from Cm into an algebraic variety V intersecting a finite set of fixed hypersurfaces in subgeneral position. As an application, we prove a result on algebraically degenerate of holomorphic curves intersecting hypersurfaces and difference analogue of Picard's theorem on holomorphic curves. Furthermore, we obtain a second main theorem of meromorphic mappings intersecting hypersurfaces in N-subgeneral position for Veronese embedding in Pn(C) and a uniqueness theorem sharing hypersurfaces.
Minor corrections, to appear in Annali SNS.
Let $c\in \mathbb{C}^{m},$ $f:\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over the field $\mathcal{P}_{c}$ of $c$-periodic meromorphic functions in $\mathbb{C}^{m}$, and let $H_{j}$ $(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position of $\mathbb{P}^{n}(\mathbb{C}).$ We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of $f$ instead of its Wronskian determinant. As its applications, we obt…
A uniqueness theorem for meromorphic maps into $\mathbb{P}^n$ with generic $(2n+2)$ hyperplanes
本文证明了带有一般 $(2n+2)$ 超平面且代数非退化的映入 $\mathbb{P}^n$ 的亚纯映射的唯一性定理，并给出了该结果的完整证明。
Meromorphic mappings on Kähler manifolds weakly sharing hyperplanes in $\mathbb P^n(\mathbb C)$
本文研究了Kähler流形到复射影空间的Meromorphic映射在共享超平面条件下的唯一性问题，并改进了相关代数依赖问题。
Let $M$ be a generic CR submanifold in $\C^{m+n}$, $m= CRdim M \geq 1$,$n=codim M \geq 1$, $d=dim M = 2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\cal D}_f, [Γ_f])$, where: 1. $f: {\cal D}_f \to Y$ is a ${\cal C}^1$-smooth mapping defined over a dense open subset ${\cal D}_f$ of $M$ with values in a projective manifold $Y$; 2. The closure $Γ_f$ of its graph in $\C^{m+n} \times Y$ defines a oriented scarred ${\cal C}^1$-smooth CR manifold of CR dimension $m$ (i.e. CR outside a closed thin set) and 3. Such that $d[Γ_f]=0$ in the sense of currents. We prove in…
Power of a meromorphic function sharing value with its k-th order directional derivative in C^m
本文研究了多复变 C^m 中亚纯函数的幂与其 k 阶方向导数共享值的唯一性问题，将单变量唯一性定理推广至高维，并通过实例证明结果的最佳性。
Second main theorem and unicity of meromorphic mappings for hypersurfaces of projective varieties in subgeneral position
本文证明了截断计数函数下到射影簇的亚纯映射的次主定理，以及不计重数共享少数超曲面映射的唯一性定理。
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