Kai Liu

Nanchang University · Department of mathematics

The fact that the complex differential polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)^{n}f'(z)-a$$\end{document} has infinitely many zeros whenever a i...

Fermat type functional equation with four terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(z)^{n}+g(z)^{n}+h(z)^{n}+k(z)^{n}=1 \end{aligned}$$\en...

We study properties of transcendental meromorphic solutions of crossed complex differential systems of equations. For instance, we study the crossed Riccati differential system f (z) 2 = 1 − g ′ (z), g(z) 2 = 1 − f ′ (z), and the crossed Weierstrass differential system f (z) 3 = 1 − g ′ (z) 2 , g(z) 3 = 1 − f ′ (z) 2. In addition, we establish a cr...

This paper is to establish new results on the zeros of \(F^{(k)}-\alpha (z)\), where F(z) is a differential polynomial or difference polynomial of f and \(\alpha (z)\) is a small function with respect to f in the sense of Nevanlinna theory. We also obtain that at least one of \(F^{(k)}-\alpha (z)\) and \(G^{(k)}-\alpha (z)\) has infinitely many zer...

This paper is devoted to considering the quasiperiodicity of complex differential polynomials, complex difference polynomials and complex delay-differential polynomials of certain types, and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.

We give the alternative proofs to consider Fermat systems of complex differential or difference or delay-differential equations. In addition, we also use value distribution of meromorphic functions to consider the existence of meromorphic solutions of complex differential or delay-differential systems.

In this paper, we mainly consider the Fermat type differential equation f (z) n + f ′ (z) n = φ(z), where φ(z) = e h(z) or 1 − e 2h(z) , and h(z) is any entire function, and the Fermat type difference equation f (z) n + f (z + c) m = e P (z) , where P (z) is any entire function and c is a non-zero constant. We also provide short proofs for some exi...

The systems of nonlinear differential equations of certain types can be simplified to matrix forms. Two types of matrix differential equations will be considered in the paper, one is Fermat type matrix differential equation A(z)n+A′(z)n=E where n = 2 and n = 3, another is Malmquist type matrix differential equation A′(z)=αA(z)2+βA(z)+γE, where α...

This paper concerns value cross-sharing of meromorphic functions, which is a variation to consider the uniqueness theory of meromorphic functions as usual. For example, we consider the uniqueness problems related to \(f(z)^n\) and \(g'(z)\) share common values together with \(g(z)^n\) and \(f'(z)\) share common values, or \(f(z)^n\) and \(g(z+c)^{m...

The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, f(z)nf(k)(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document...

By observing the periodicity of transcendental entire solutions of the complex differential equation f(z)f″(z)=p(z)sin2⁡z, where p(z) is a non-zero polynomial with real coefficients and real zeros, Yang's Conjecture has been proposed and considered by many authors recently. In this paper, we consider the parity of transcendental entire solutions of...

In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of f (z) n L(g) − a(z) and g(z) n L(f) − a(z), where L(h) takes the derivatives h (k) (z) or the shift h(z+c) or the difference h(z+c)−h(z) or the delay-differential h (k) (z+c), where k is a positive integer, c is a non-zero constant and a...

We discuss the relationship on the periodicity of a transcendental entire function with its differential polynomials. For example, we obtain that if f is a transcendental entire function, k is a non-negative integer and if (anfn + ⋯ + a1f)(k) is a periodic function, then f is also a periodic function, where a1, … an (≠ 0) are constants. Our results...

This book presents developments and new results on complex differential-difference equations, an area with important and interesting applications, which also gathers increasing attention. Key problems, methods, and results related to complex differential-difference equations are collected to offer an up-to-date overview of the field. Presents recen...

We investigate the existence of non-trivial holomorphic and meromorphic solutions of Fermat functional equations over an open Riemann surface $S$. When $S$ is hyperbolic, we prove that any $k$-term Fermat functional equation always exists non-trivial holomorphic and meromorphic solution. When $S$ is a general open Riemann surface, we prove that eve...

Steinmetz [16] considered the first order non-linear differential equations $$C(z, f)(f^\prime)^2+B(z, f)f^\prime+A(z, f)=0,$$ where A(z, f), B(z, f), C(z, f) are polynomials in f with rational coefficients in z and pointed out that the above equation must reduce into some certain types when it admits transcendental meromorphic solutions. In this p...

Abstract This paper is devoted to the uniqueness of q-difference-differential polynomials of different types. Using the idea of common zeros and common poles (Chin. Ann. Math., Ser. A 35:675–684, 2014), we improve the conditions of the former theorems and obtain some new results on the uniqueness of q-difference-differential polynomials of meromorp...

Exponential type functions are important subclasses of transcendental entire functions. In this paper, we will use some results given by Steinmetz (Manuscr Math 26:155–167, 1978) to consider the zeros of difference or differential-difference polynomials of exponential polynomials. In addition, we also consider the zeros of difference polynomials of...

We obtain necessary conditions for the non-linear complex differential-difference equations w(z+1)w(z−1)+a(z)w′(z)w(z)=R(z,w(z)) to admit transcendental meromorphic solutions w(z) such that ρ2(w) < 1, where R(z,w(z)) is rational in w(z) with rational coefficients, a(z) is a rational function and ρ2(w) is the hyper-order of w(z). Our results can be...

We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value.

In this paper, we give some necessary conditions on the existence of meromorphic solutions on Fermat type difference equations. We also consider the properties of transcendental entire solutions on the systems of Fermat type differential-difference equations.

Abstract In this paper, we consider the zeros distribution of f(z)P(z,f)−q(z) $f(z)P(z,f) -q(z)$, where P(z,f) $P(z,f)$ is a linear differential-difference polynomial of a finite-order transcendental entire function f(z) $f(z)$, and q(z) $q(z)$ is a nonzero polynomial. To a certain extent, Theorem 1.1 generalizes the recent results (Latreuch and Be...

We consider the non-linear differential-difference equation $$ c(z)w(z+1)+a(z)\frac{w'(z)}{w(z)}=R(z,w(z)), $$ where R(z,w(z)) is rational in w(z) with rational coefficients, a(z) and c(z) are non-zero rational functions. We give necessary conditions on the degree of $R(z,w)$ for the above equation to admit a transcendental meromorphic solution of...

This paper is to consider the generalized Fermat difference equations with different types which ever considered by Li [14], Ishizaki and Korhonen [9], Zhang [26] and Liu [15–18], respectively. Some new observations and results on these equations will be given.

This paper is devoted to exploring the properties of meromorphic solutions on complex differential–difference equations using Nevanlinna theory. We state some relationships between the exponent of convergence of zeros with the order of meromorphic solutions on linear or non-linear differential–difference equations.

The Valiron–Mohon’ko theorem plays an important role in the theory of complex differential equations. In this paper, a difference analogue of the Valiron–Mohon’ko theorem is established, which can be used to get the characteristic functions on irreducible rational functions of f(z) and its shifts. Using our results and some properties of periodic f...

The main aim of the paper is to improve some classical results on the distribution of zeros for differential polynomials and differential-difference polynomials. We present some results on the distribution of zeros of [f(z)ⁿf(z + c)](k) − α(z) and [f(z)ⁿ(f(z + c) − f(z))](k) − α(z) and give some examples to show that the results are best possible i...

In this paper, we will consider value sharing problems on a transcendental entire function f with its differential-difference polynomials. We establish some results that can be viewed as the differential-difference analogues of Brück conjecture.

In this article, we investigate the uniqueness problem on a transcendental entire function f⁢(z) with its linear mixed-operators Tf, where T is a linear combination of differential-difference operators Dην:=f(ν)⁢(z+η) and shift operators Eζ:=f⁢(z+ζ), where η,ν,ζ are constants. We obtain that if a transcendental entire function f⁢(z) satisfies λ⁢(f-...

In this paper, we mainly deal with the problem that f(qz) and f’(z) share common values. One of the purpose is to explore whether the classical uniqueness results remain valid or not by considering some uniqueness theorems on f(qz) and f’(z) sharing common values. Some examples and remarks are given to show that our results are sharp in certain sen...

We consider the exponential polynomials solutions of non-linear differential-difference equation \({f(z)^{n}+q(z)e^{Q(z)}f^{(k)}(z+c) = P(z)}\), where q(z), Q(z), P(z) are polynomials and n, k are positive integers and the linear differential-difference equation \({f'(z) = f(z + c)}\). Our results show that any exponential polynomials’ solutions of...

In this paper, we investigate the uniqueness problem of a meromorphic function sharing one small function with its differential polynomial, and give a result which is related to a conjecture of R. Brück.

Tropical Nevanlinna theory studies value distribution of continuous piecewise linear functions of a real variable. In this paper, we use the reasoning from tropical Nevanlinna theory to present tropical counterparts of some classical complex results related to Fermat type equations, Hayman conjecture and Bruck conjecture.

The purpose of this paper is to describe the meromorphic solutions of Fermat type equations. By considering the existence and growth of meromorphic solutions, we explore the similarities and differences between Fermat types differential equations with difference equations. Several non-entire meromorphic solutions of some equations with certain type...

In this paper, we deal with value distribution for q-shift polynomials of tran-scendental meromorphic functions with zero order and obtain some results which improve the previous theorems given by Liu and Qi [18]. In addition, we investigate value sharing for q-shift polynomials of transcendental entire functions with zero order and obtain some res...

This article we explore the relationship between the number of differential and difference operators with the existence of meromorphic solutions of Fermat type differential and difference equations. Some Fermat differential and difference equations of certain types are also considered.

In this article, we study entire solutions of the nonlinear differential-difference equation $$ q(z)f^{n}(z)+a(z)f^{(k)}(z+1)=p_1(z)e^{q_1(z)}+p_2(z)e^{q_2(z)} $$ where $p_1(z)$, $p_2(z)$ are nonzero polynomials, $q_1(z)$, $q_2(z)$ are nonconstant polynomials, $q(z)$, $a(z)$ are nonzero entire functions of finite order, $n\geq2$ is an integer. We...

In this paper, we investigate the value distribution of q-difference polynomials of meromorphic function of finite logarithmic order, and study the zero distribution of difference-differential polynomials [f n (z)f(qz + c)](k) and [f n (z)(f(qz + c)−f(z))](k), where f(z) is a transcendental function of zero order. The uniqueness problem of differ...

In this paper, we consider the growth and existence of solutions of differential-difference equations of certain types. We also consider the differential-difference analogues of Brück conjecture and give a short proof on a theorem given by Li, Yang and Yi [18]. Our additional purpose is to explore the similarity or difference on some problems in di...

This paper is devoted to considering the zeros of complex differential-difference polynomials of different types. Our results can be seen as the differential-difference analogues of Hayman conjecture (Ann. Math. 70:9-42, 1959). MSC: 30D35, 39A05.

For a complex value q ≠ 0 , 1 , and a transcendental entire function f ( z ) with order, 0 < σ ( f ) < ∞ , we study the value distribution of q -difference differential polynomials [ f n ( z ) ( f ( q z ) − f ( z ) ) ] ( k ) and [ f ( z ) f ( q z ) ] ( k ) . MSC: 30D35, 39A05.

In this paper, we will investigate the properties of entire solutions with finite order of the Fermat type difference or differential-difference equations. This is continuation of a recent paper (Liu et al. in Arch. Math. 99, 147-155, 2012). In addition, we also consider the value distribution and growth of the entire solutions of linear differenti...

The concept of logarithmic order is used to investigate the growth of solutions of the linear differential equations f(k) + Ak-1 +...A1(z)f' + A0(z)f= 0, f(k) + A k-1 +...A1(z)f' + A0(z)f= F(z),where A 0 ≠ 0, A1 ..., Ak-1 and F ≠ 0 are transcendental entire functions with orders zero.

In this article, we describe the finite-order transcendental entire solutions of Fermat type q-difference and q-difference differential equations. In addition, we investigate the similarities and other properties among those solutions.

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.

We consider the zero distribution of difference-differential polynomials of meromorphic functions and present some results which can be seen as the discrete analogues of the Hayman conjecture. In addition, we also investigate the uniqueness of difference-differential polynomials of entire functions sharing one common value. Our theorems improve som...

In this paper, we show that for any finite order entire function f(z), the function of the form f(z)n[f(z+c)-f(z)]s has no nonzero finite Picard exceptional value for all nonnegative integers n, s satisfying n ≥ 3, which can be viewed as a difference result on Hayman conjecture. We also obtain some uniqueness theorems for difference polynomials of...

In this paper, the zero distribution of differential-difference polynomials and will be considered. The results can be seen as the differential-difference analogues of Hayman conjecture. MSC: 30D35, 39A05.

We mainly discuss entire solutions with finite order of the following Fermat type differential-difference equations $$\begin{array}{ll}(f)^{n}+f(z+c)^{m}=1;\\f^{\prime}(z)^{n}+f(z+c)^{m}=1;\\ f^{\prime}(z)^{n}+[f(z+c)-f(z)]^{m}=1,\end{array}$$ where m, n are positive integers.

We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions. We also investigate the uniqueness problems of difference-differential polynomials of entire functions sharing a common value.

In this paper, we investigate sharing value problems related to a meromorphic function f(z) and f(qz), where q is a non-zero constant. It is shown, for instance, that if f(z) is zero-order and shares two valves CM and one value IM with f(qz), then f(z) = f(qz).

In this paper, we investigate the value distribution of differences of meromorphic functions. Some results are proved concerning the existence of zeros of the fκΔcf - a(z), κ ε Z, which can be viewed as discrete analogues of the Hayman conjecture [10].

The main purpose of this paper is to consider the differential equation u((m)) = P (z)u (m >= 2) where P is a polynomial with complex, in general, coefficients. Let z(k)(u), k = 1, 2, ... be the zeros of a nonzero solution u to that equation. We obtain bounds for the sums Sigma(j)(k=1) 1/vertical bar z(k)(u)vertical bar (j is an element of N) which...

The main purpose of this paper is to consider the differential equation $u^{(m)}=P(z)u$ $(m\geq 2)$ where $P$ is a polynomial with in general complex coefficients. Let $z_{k}(u),$ $k=1,2,\ldots$ be the zeros of a nonzero solution $u$ to that equation. We obtain bounds for the sums $$\sum_{k=1}^{j}\frac{1}{|z_{k}(u)|}\quad (j\in\mathbb{N})$$ which e...

In this paper, we consider the zero distributions of q-shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to q-shift difference polynomials. We also investigate the uniqueness problem of q-shift difference polynomials that s...

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 poly...

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of fn(z)(λfm(z+c)+μfm(z))−a, where f is an entire function, n, m are two i...

In this paper we obtain that there are no transcendental entire solutions with finite order of some nonlinear difference equations of different forms.

The main purpose of this paper is to investigate the oscillation theory of meromorphic solutions of the second order linear differential equation f″+A(z)f=0 for the case where A is meromorphic in the unit disc D={z:|z|1}.

We investigate the zero distributions of difference polynomials of meromorphic functions, which can be viewed as the Hayman conjecture (introduced by Hayman in 1967) for differences. And we also study the uniqueness of difference polynomials of meromorphic functions sharing a common value, and obtain uniqueness theorems for differences.

We establish a q-shift difference analogue of the logarithmic derivative lemma. We also investigate the value distributions of q-shift difference polynomials and the growth of solutions of complex q-shift difference equations.

In this paper, we investigate the uniqueness problems of difference polynomials of meromorphic functions that share a value or a fixed point. We also obtain several results concerning the shifts of meromorphic functions and the sufficient conditions for periodicity which improve some recent results in Heittokangas et al. (2009) [10] and Liu (2009)...

This research is a continuation of a recent paper, due to Liu and Laine, dealing with difference polynomials of entire function. In this paper, we investigate the value distribution of difference polynomials of meromorphic functions and prove some difference analogues to some classical results for differential polynomials. Mathematics Subject Clas...

In this paper, we investigate the value distribution of difference polynomials and prove some difference analogues of results of Hayman and the Brück conjecture.

We have two purposes. First, we prove two theorems and two corollaries on normal families which improve and generalize some results of X. Pang and L. Zalcman [Bull. Lond. Math. Soc. 32, No.3, 325–331 (2000; Zbl 1030.30031)], G. Zhang, W. Sun and X. Pang [Chin. Ann. Math., Ser. A 26, No. 6, 765–770 (2005; Zbl 1098.30028)], and J. Chang and M. Fang [...

This paper is devoted to proving some uniqueness type results for an entire function f(z) that shares a common set with its shift f(z+c) or its difference operator Δcf. We also give some applications to solutions of non-linear difference equations related to a conjecture proposed by C.C. Yang.

In this paper, we investigate the growth of solutions and the existence of subnormal solutions for a class of higher order linear differential equations. We obtain some results which improve and extend the results of Chen-Shon [2] and Gundersen-Steinbart [5].

In this paper, we deal with the problem of uniqueness of meromorphic functions that share two small functions with their derivatives, and obtain the following result, which improves a result of W. H. Yao and P. Li [J. Math. Anal. Appl. 322, No. 1, 133–145 (2006; Zbl 1101.30032)]: Let f(z) be a non-constant meromorphic function and k>5 an integer. I...

We investigate the value distribution of difference operators for entire functions, and establish a difference analogue of the Brück conjecture for entire functions of order less than 1.

In this paper, we deal with the problem of the existence of entire solutions of certain type differential equations using Nevanlinna theory, and answer a question proposed by C. C. Yang.

In this paper, we deal with the problems of uniqueness of meromorphic functions that share one finite value with their derivatives and obtain some results that improve the results given by Rainer Brück and Qingcai Zhang.