Walter Bergweiler, J. K. Langley
Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = f(z+1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f.
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The growth of meromorphic solutions of linear difference equations containing Askey-Wilson divided difference operators is estimated. The $φ$-order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order $ρ_{\log}(f)$ and the classical order $ρ(f)$ of a meromorphic function $f$.
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.
In this paper we investigate the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations with entire or meromorphic coefficients. We further extend and improve few results on the order of meromorphic solutions by using (p,q)-lower order and (p,q)-lower type followed by the investigation of Luo and Zheng (2016), Belaidi and Bellaama (2020).
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 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.
In this paper we have studied the growth of meromorphic solutions of higher order homogeneous and non-homogeneous linear difference equations with entire and meromorphic coefficients. We have extended and improved some results of Zhou and Zheng (2017), Belaidi and Benkarouba (2019) by using (p,q)-order and (p,q)-type.
In this paper, we give a simple proof and strengthening of a uniqueness theorem of meromorphic functions which partially share 0, $\infty$ CM and 1 IM with their difference operators. Meanwhile, we partial solve a conjecture given by Chen-Yi in \cite{CY} and generalize some previous theorems in \cite{C, CX}.
We introduce and develop the root locus method in mathematics. And we study the distribution of zeros of meromorphic functions by root locus method.
The paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.
In this article we have studied complex linear homogeneous difference equations where the coefficients are meromorphic functions, having finite iterated p-phi order. We have made some estimations on the growth of its nontrivial solutions. Also we have extended some of the previous results of Zhou-Zheng (2017), Belaïdi-Benkarouba (2019) and Belaïdi-Bellaama (2021).
In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some inequalities for the relationship of the zero counting function of $f(z)^nH(z, f)-s(z)$ and the characteristic function and pole counting function of $f(z)$. Based on these inequalities, we establish some difference analogues of a classical result of Hayman for meromorphic functions. Some special cases are also investigated. These results improve previous findin…
It is shown that if It is shown that if \begin{equation}\label{abstract_eq} f(z+1)^n=R(z,f),\tag† \end{equation} where $R(z,f)$ is rational in $f$ with meromorphic coefficients and $°_f(R(z,f))=n$, has an admissible meromorphic solution, then either $f$ satisfies a difference linear or Riccati equation with meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if \eqref{abstract_eq}, where the assumption $°_f(R(z,f))=n$ has been discarded, has rational coefficients and a transce…
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.
The $φ$-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of meromorphic functions in the complex plane are investigated in terms of the $φ$-order, which measures the growth of functions between the classical order and the logarithmic order. Several results on value distribution of meromorphic functions are discussed by using the $φ$-order and the $φ$-exponent of convergence. Instead of linear differential equations, the applications in the complex plane lie i…
A number of results are proved concerning the existence of non-real zeros of derivatives of strictly non-real meromorphic functions in the plane.
A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane
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 study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations.
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.
Meromorphic solutions of linear $q$-difference equations
本文在复域中构造了一阶线性 $q$-差分方程的显式亚纯解，描述了零点和极点位置，并利用因式分解推广至高阶情形。
The paper determines all meromorphic functions with finitely many zeros in the plane having the property that a linear differential polynomial in the function, of order at least 3 and with rational functions as coefficients, also has finitely many zeros.
On meromorphic solutions of certain Fermat-type difference and analogues equations concerning open problems
本文研究某些Fermat型移位与差分方程的亚纯解，证明其可由Riccati型函数生成，并解决了Liu和Yang提出的未决问题，同时用实例验证了相关结论。
Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple exept finitely many and T(r,h)=o{T(r,f)} as r tends to infinity, then f'=h has infinitely many solutions (including poles).
We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the case of meromorphic functions with finitely many poles. We show by examples the precision of our results. One of our main tools is the Fatou theorem from complex dynamics.
In this article, we deal with the order of growth of solutions of non-homogeneous linear differential-difference equation \begin{equation*} \sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F(z), \end{equation*} where $A_{ij},$ $F\left( z\right) $ are entire or meromorphic functions and $c_{i}$ $\left( 0,1,...,n\right) $ are non-zero distinct complex numbers. Under the sufficient condition that there exists one coefficient having the maximal lower order or having the maximal lower type strictly greater than the order or the type of other coefficients, we obtain estimates of the lower bound of…
The zeta function of zeros and poles of a meromorphic function, the criterion for the absence of zeros, and its application to sums of Cauchy kernels
该研究通过联系亚纯函数零点与极点的 zeta 函数，提出零点不存在判据，并给出 Cauchy 核之和零点存在的充分条件。
Meromorphic Solutions of Difference Equations Involving Borel and Nevanlinna Exceptional Values
本文研究了差分方程中亚纯函数与其线性差分算子共享值的问题，证明了差分方程L_c^n(f)≡Af的亚纯解存在并给出了显式解，同时探讨了二阶差分方程的有理和超越亚纯解的存在性。
It is shown that the difference equation \begin{equation}\label{abseq} (Δf(z))^2=A(z)(f(z)f(z+1)-B(z)), \qquad\qquad (1) \end{equation} where $A(z)$ and $B(z)$ are meromorphic functions, possesses a continuous limit to the differential equation \begin{equation}\label{abseq2} (w')^2=A(z)(w^2-1),\qquad\qquad (2) \end{equation} which extends to solutions in certain cases. In addition, if (1) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost same in the sense of Nevanlinna under so…
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。
It is shown that if three distinct values of a meromorphic function f:C^n -> P^1 of hyper-order strictly less than 2/3 have forward invariant pre-images with respect to a translation t:C^n -> C^n, t(z)=z+c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green's Picard-type theorem in the special case where the hyper-order of f is less than 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the lemma on the logarithmic derivative and of the second …
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。
A result is proved concerning meromorphic functions of finite order in the plane such that all but finitely many zeros of the second derivative are zeros of the first derivative.
A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen \cite{AHH}, \cite {HK-2} that a connection exists between discrete Painlevé equations and (finite order) Nevanlinna theory is an estimate of the integrated average of $\log^+|f(z+1)/f(z)|$ on $|z|=r$. We obtained essentially the same estimate in our previous paper \cite{CF} independent of Halburd et al \cite{HK-1}. We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic…
Unicity on meromorphic function sharing three small functions CM with its higher-order difference operators
本文证明在二阶增长级小于1的条件下，若亚纯函数与其高阶差分算子共享三个互不相同的小函数CM，则二者恒等，从而建立了差分情形下的唯一性定理。
Non-real zeros of linear differential polynomials in real meromorphic functions
该研究探讨实整函数相关微分多项式的零点分布问题，证明了仅具实零点的无限增长阶实整函数（或其倒数）对应的f''+ωf对任意正ω有无穷多非实零点。
The maximum of the modulus of a meromorphic function cannot be restricted from above by the Nevanlinna characteristic of this meromorphic function. But integrals from the logarithm of the module of a meromorphic function allow similar restrictions from above. This is illustrated by one of the important theorems of Rolf Nevanlinna in the classical monograph by A.A. Goldberg and I.V. Ostrovskii on meromorphic functions, as well as by the Edrei-Fuchs Lemma on small arcs and its versions for small intervals in articles by A.F. Grishin, M.L. Sodin, T.I. Malyutina. Similar results for integrals of d…
Two meromorphic functions $ f $ and $ g $ are said to share a value $ s\in\mathbb{C}\cup\{\infty\} $ $ CM $ $ (IM) $ provided that $ f(z)-s $ and $ g(z)-s $ have the same set of zeros counting multiplicities (ignoring multiplicities). We say that a meromorphic function $ f $ share $ s\in\mathbb{C} $ partially $ CM $ with a meromorphic function $ g $ if $ E(s,f)\subseteq E(s,g)$. It is easy to see that the condition ``partially shared values $ CM $" is more general than the condition ``shared value $ CM $". With the idea of partially shared values, in this paper, we prove some uniqueness result…
Zero order meromorphic solutions of $q$-difference equations of Malmquist type
本文研究 Malmquist 型 q-差分方程 f(qz)^n=R(z,f)，证明当 |q|≠1 时其零级超越亚纯解仅存在于可约化为 q-差分线性或 Riccati 方程的方程中，并给出自治情形的显式解及差分 Painlevé 性质的应用。
All meromorphic solutions of Fermat-type functional equations
本文利用椭函数性质，研究了复平面上费马型函数方程f(z)^n + f(L(z))^m = 1的亚纯解，并探讨了差分和q差分方程的亚纯解。
Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with the maximum function of its modulus $M(r,f)$ on circles centered at zero of radius $r$. A number of classical, well-known and widely used results allow us to estimate from above the integrals of the positive part of the logarithm $\ln^+M(t,f)$ over subsets of $E\subset [0,r]$ via the Nevanlinna characteristic $T(r,f)$ and the linear Lebesgue measure of the set $E$. The paper gives similar estimates for the Lebesgue-Stieltjes integrals of $\ln^+M (t,f)$ over the increasing integration function of $m$. The main part of the p…
Zeros of meromorphic functions of the form $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$
本文研究了形如 $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$ 的亚纯函数的零点分布，证明了经典 Keldysh 定理的类比，并揭示了此类无零点函数与多项式系数二阶微分方程之间的关系。
In this paper, we mainly propose improvements of the logarithmic difference lemma for meromorphic functions in several complex variables, and then investigate meromorphic solutions of partial difference equations from the viewpoint of Nevanlinna theory.
Let w(z) be a finite-order meromorphic solution of the second-order difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else equation (1) can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painleve equation of the form (1), together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.
The main result of the paper determines all real meromorphic functions of finite order in the plane for which the first derivative has finitely many zeros, while the function itself and one of its higher derivatives have finitely many non-real zeros.
Let $U\not\equiv \pm\infty$ be a $δ$-subharmonic function on a closed disc of radius $R$ centered at zero. In the previous two parts of our paper, we obtained general and explicit estimates of the integral of the positive part of the radial maximum growth characteristic ${\mathsf M}_U(t):= \sup\bigl\{U(z)\bigm| |z|=r\bigr\}$ over the increasing integration function $m$ on the segment $[0, r]\subset [0,R)$ through the difference characteristic of Nevanlinna and the quantities associated with the integration function $m$. The third part of our paper contains estimates of these quantities in term…
By extending the idea of a difference operator with a fixed step to varying-steps difference operators, we have established a difference Nevanlinna theory for meromorphic functions with the steps tending to zero (vanishing period) and a difference Nevanlinna theory for finite order meromorphic functions with the steps tending to infinity (infinite period) in this paper. We can recover the classical little Picard theorem from the vanishing period theory, but we require additional finite order growth restriction for meromorphic functions from the infinite period theory. Then we give some applica…
Let $U\not\equiv \pm\infty$ be the difference of subharmonic functions, i.e., a $δ$-subharmonic function, on a closed disc of radius $R$ centered at zero. In the preceding first part of our paper, we obtained general estimates for the integral of the positive part of the radial maximum growth characteristic ${\mathsf M}_U(t):=\sup\bigl\{U(z)\bigm| |z|=r\bigr\}$ over the increasing integration function $m$ on the segment $[0, r]$ via the Nevanlinna difference characteristic and the modulus of continuity of the function $m$. The second part of the work gives an explicit view for such estimates, …
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…
Let h be a complex meromorphic function decomposed in two different ways P(f) and Q(g), where f, g are meromorphic functions and P, Q are rational functions. We follow an approach due to C.-C. Yang, P. Li and K. H. Ha who handle similar decompositions, however, with P, Q polynomials, by applying the Second Nevanlinna Theorem. We establish relations between the number k of distinct zeros of P', the degrees of P and Q, and (unless f, g are analytic functions) the number of distinct zeros of the denominators of P and Q.
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。
Subnormal transcendental meromorphic solutions of difference equations with Schwarzian derivative
本文利用 Nevanlinna 理论研究了三个含 Schwarzian 导数的差分方程，获得了其次正规超越亚纯解存在的必要条件，并给出实例支持。
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.
Let D be the open unit disc in C. The paper deals with the following conjecture: If f is a continuous function on bD such that the change of argument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1 has no zero on bD then f extends holomorphically through D. We prove a related result on meromorphic extendibility for smooth functions with finitely many zeros of finite order, which, in particular, implies that the conjecture holds for real analytic functions.
The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By using the theorem, we reprove some results about zeros of derivative of Xi function,Gamma function and digamma function in a new way.
After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace transformations of convenient orders and convenient direction in order to construct a solution of the same equation that is meromorphic on $\mathbb{C}^{*}$. We use this theorem to construct explicitly an invertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on $\mathbb{C}^{*}$. Moreover, when the system is…
The HRT conjecture for a class of meromorphic functions
本文证明非零亚纯函数的有限时频平移系统线性无关，从而证实HRT猜想对可延拓为除有限点外解析的平方可积函数成立。
On Meromorphic Solutions to a Difference Equation of Tumura-Clunie Type
本文利用Nevanlinna理论刻画了一类Tumura-Clunie型非线性差分方程在ρ_2(f)<1条件下的亚纯解，改进了Chen等人的已有结果，并推广到微分-差分多项式情形。
We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng Problem.
We consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. We show that such functions exist if and only if the rays are equally spaced. We also obtain a normal family analogue of this result.
The present paper deals with the convergence properties of multi-level Hermite-Padé approximants for a class of meromorphic functions given by rational perturbations with real coefficients of a Nikishin system of functions, and study the zero location of the corresponding multiple orthogonal polynomials.
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