In this paper, we study the uniqueness of the difference of meromorphic functions. We prove the following result: Let $f$ be a non-constant meromorphic function of hyper-order less than $1$, let $η$ be a non-zero complex number, $n\geq1$, an integer, and let $a,b,c\in\hat{S}(f)$ be three distinct small functions and two of them be periodic small functions with period $η$. If $f$ and $Δ_η^{n}f$ share $a,b,c$ CM, then $f\equivΔ_η^{n}f$.
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.
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.
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.
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.
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.
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].
Two meromorphic functions $f(z)$ and $g(z)$ sharing a small function $α(z)$ usually is defined in terms of vanishing of the functions $f-α$ and $g-α$. We argue that it would be better to modify this definition at the points where $α$ has poles. Related to this issue we also point out some possible gaps in proofs in the published literature.
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 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.
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…
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.
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…
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.
The paper is devoted to study the uniqueness problem of linear delay-differential operator of a meromorphic function sharing two sets or small function together with values with its $c$-shift and $q$-shift operator. Results of this paper drastically improve two recent results of Meng-Liu [J. Appl. Math. Inform. 37(1-2)(2019), 133-148] and Qi-Li-Yang [Comput. Methods Funct. Theo., 18(2018), 567-582]. In addition to this, one of our result improve and extend that of Qi-Yang [Comput. Methods Funct. Theo., 20(2020), 159-178].
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…
Let F and G be two families of meromorphic functions on a domain D, and let a, b and c be three distinct points in the extended complex plane. Let G be a normal family in D such that all limit functions of G are non-constant. If for each f in F, there exists g in G such that f and g share a, b and c partially, then F is normal in D. This gives a sharp improvement of a result due to X. J. Liu, S. H. Li and X. C. Pang. We also prove some interesting related sharp results.
In this paper we prove the result: Let $\mathcal{F}$ be a family of meromorphic functions on a domain $Ω$ such that every pair of members of $\mathcal{F}$ shares a set $S:=\left\{ψ_1(z), ψ_2(z), ψ_3(z) \right\}$ in $Ω$, where $ψ_j(z), \ j=1,2,3$ is meromorphic in $Ω.$ If for every $f\in \mathcal{F}$, $f(z_0)\neq ψ_i (z_0)$ whenever $ψ_i(z_0)=ψ_j(z_0)$ for $i,j\in \left\{1,2,3 \right\}(i\neq j)$ and $z_0\in Ω,$ then $\mathcal{F}$ is normal in $Ω$. This result generalizes a result of M.Fang and W.Hong [Some results on normal family of meromorphic functions, Bull. Malays. Math. Sci. Soc. (2)23 (2…
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing $q\ (q> C+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $C <2n$ (explicitly estimated), then there are some algebraic relation between them. A degeneracy theorem for $k\ (2\le k\le n+1)$ meromorphic mappings sharing hyperp…
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…
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing $q\ (q> 2n+1+α+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $α<1$ (explicitly estimated), then $f^1=f^2$ or $f^2=f^3$ or $f^3=f^1$. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to l…
An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $πi$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k-th$ derivatives. We improves the author's result \cite{h} from entire function to meromorphic function, the first derivative to its differential-difference polynomial, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $ρ_{2}(f)<1$, l…
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.
The theory of strong uniqueness polynomials, satisfying the separation condition (first introduced by Fujimoto \cite{Fuj1}), for complex meromorphic functions is quite complete. We construct examples of strong uniqueness polynomials which do not necessary satisfy the separation condition by constructing regular 1-forms of Wronskian type, a method introduced in \cite{AWW}. We also use this method to produce a much easier proof in establishing the necessary and sufficient conditions for a polynomial, satisfying the separation condition, to be a strong uniqueness polynomials for meromorphic funct…
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…
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.
