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 functions and rational functions.
We prove some uniqueness results which improve and generalize results of Jiang-Tao Li and Ping Li[Uniqueness of entire functions concerning differential polynomials. Commun. Korean Math. Soc. 30 (2015), No. 2, pp. 93-101].
In this paper we shall find some sufficient conditions for a uniqueness polynomial to be a strong uniqueness polynomial, as this type of problem was never investigated by researchers earlier.We also exhibit some examples to substantiate our theorems.
The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results have been obtained. In this paper, we study a transcendental entire function f that shares a non-zero polynomial a with f', together with its linear differential polynomials of the form with rational function coefficients.
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 E be a compact set of C of positive logarithmic capacity. Let us suppose that for every polynomial $P\not=id$ we have $P^{-1}(E)\not=E$. Then for all no constant polynomials f and g such that $f^{-1}(E)=g^{-1}(E)$ we have f=g.
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 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 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 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].
We consider the zeros distributions on the derivatives of difference polynomials of meromorphic functions, and present some results which can be seen as the discrete analogues of Hayman conjecture \cite{hayman1}, also partly answer the question given in \cite[P448]{luolin}. We also investigate the uniqueness problems of difference-differential polynomials of entire functions sharing one common value. These theorems improve the results of Luo and Lin\cite{luolin} and some results of present authors \cite{liuliucao}.
In this paper, we continue to investigate the uniqueness problem when an entire function $f$ and its linear differential polynomial $L(f)$ share two distinct complex values CMW (counting multiplicities in the weak sense) jointly. Also, We investigate the same problem when $f$ and its differential monomial $M(f)$ share two distinct complex values CMW. Our results generalize the recent result of Lahiri (Comput. Methods Funct. Theory, https://doi.org/10.1007/s40315-020-00355-4).
In this paper, we investigate the uniqueness problem of difference polynomials $f^{n}(z)P(f(z))L_c(f)$ and $g^{n}(z)P(g(z))L_c(g)$, where $L_c(f)=f(z+c)+c_0f(z)$, $P(z)$ is a polynomial with constant coefficients of degree $m$ sharing a small function with the notions of weakly weighted sharing and relaxed weighted sharing and obtained the corresponding results, which improve and extend some recent results due to Sahoo and Biswas (Tamkang J. Math., \textbf{49}(2), 85--97 (2018)).
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.
We investigate the sets of uniform limits $A(\bar{B}_n)$, $A(\bar{D}^I)$ of polynomials on the closed unit ball $\bar{B}_n$ of $\mathbb{C}^n$ and on the cartesian product $\bar{D}^I$ where $I$ is an arbitrary set and $\bar{D}$ is the closed unit disc in $\mathbb{C}$. We introduce the notion of set of uniqueness for $A(\bar{D}^I)$ (respectively for $A(\bar{B}_n)$) for compact subsets $K$ of $T^I$ (respectively of $\partial \bar{B}_n$) where $T=\partial D$ is the unit circle. Our main result is that if $K$ has positive measure then $K$ is a set of uniqueness. The converse does not hold. Finally,…
In this paper we consider class of continuous functions, called quasiaharmonic functions, admitting best approximations by harmonic polynomials. In this class we prove a uniqueness theorem by analogy with the analytic functions.
Let f be a non constant meromorphic function and a(not identically zero or infinity) be a meromorphic function satisfying T(r,a) = o(T(r,f)) as r tends to infinity, and p(z) be a polynomial of degree n greater than or equal to 1 with p(0) = 0. Let P[f] be a non constant differential polynomial of f. Under certain essential conditions, we prove the uniqueness of p(f) and P[f] when p(f) and P[f] share a with weight l greater than or equal to zero. Our result generalizes the results due to Zang and Lu, Banerjee and Majumder, Bhoosnurmath and Kabbur and answers a question of Zang and Lu.
In this paper, with the help of the idea of weakly weighted sharing introduced by \emph{Lin -Lin} [Kodai Math. J., 29(2006), 269-280], we study the uniqueness of a polynomial expression $ P(f) $ and $ [P(f)]^{(k)} $ of a meromorphic function $ f $ sharing a small function. The main results significantly improved the result of \emph{Liu - Gu} [Kodai Math. J., 27(3)(2004), 272-279]. This research work explores certain condition under which the polynomial $ P(f) $ can be reduced to a non-zero monomial, and as a consequence, the specific form of the function $ f $ is obtained. By some constructive…
Let P(z) and Q(y) be polynomials of the same degree k>=1 in the complex variables z and y, respectively. In this extended abstract we study the non-linear functional equation P(z)=Q(y(z)), where y(z) is restricted to be analytic in a neighborhood of z=0. We provide sufficient conditions to ensure that all the roots of Q(y) are contained within the range of y(z) as well as to have y(z)=z as the unique analytic solution of the non-linear equation. Our results are motivated from uniqueness considerations of polynomial canonical representations of the phase or amplitude terms of oscillatory integr…
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.
Let f be a non-constant meromorphic function and a = a(z) be a small function of f. Under certain essential conditions, we obtained similar type conclusion of Bruck Conjecture, when f and its differential polynomial P[f] shares a with weight l. Our result improves and generalizes a recent result of Li, Yang, and Liu.
We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$ when $x+y=1$. A sharp estimate $d \leq 2N-3$ is known. In this paper we study the $p$ for which equality holds. We prove some new results about the form of these "sharp" polynomials. Using these new results and using two independent computational methods we give a complete classification of these polynomials up to $d=17$. The question is motivated by the probl…
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.
We obtain new uniqueness theorems for harmonic functions defined on the unit disc or in the half plane. These results are applied to obtain new resolvent descriptions of spectral subspaces of polynomially bounded groups of operators on Banach spaces.
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…
This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.
