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Let f be an entire function of finite order, let $n\geq 1$ n ≥ 1 , $m\geq 1$ m ≥ 1 , $L(z,f)\not \equiv 0$ L ( z , f ) ≢ 0 be a linear difference polynomial of f with small meromorphic coefficients, and $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 be a difference polynomial in f of degree $d\leq n-1$ d ≤ n − 1 with small meromorphic coefficients. We consider the growth and zeros of $f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$ f n ( z ) L m ( z , f ) + P d ( z , f ) . And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type $f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ f n ( z ) + P d ( z , f ) = p 1 e α 1 z + p 2 e α 2 z , where $n\geq 2$ n ≥ 2 , $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 is a difference polynomial in f of degree $d\leq n-2$ d ≤ n − 2 with small mromorphic coefficients, $p_{i}$ p i , $\alpha _{i}$ α i ($i=1,2$ i = 1 , 2 ) are nonzero constants such that $\alpha _{1}\neq \alpha _{2}$ α 1 ≠ α 2 . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

This paper is devoted to studying the growth of entire or meromorphic solutions to two systems ofq-difference differential equations. The estimates on the growth order of meromorphic solutions are obtained, which are extensions of previous results due to Xu et al . Examples are given to illustrate the existence of solutions of such systems.

In this paper, we establish certain new subclasses of meromorphic harmonic functions using the principles of q-derivative operator. We obtain new criteria of sense preserving and univalency. We also address other important aspects, such as distortion limits, preservation of convolution, and convexity limitations. Additionally, with the help of sufficiency criteria, we estimate sharp bounds of the real parts of the ratios of meromorphic harmonic functions to their sequences of partial sums.

In the paper, we introduce a new notion of reduced linear c-shift operator $L _{c}^{r}\,f$Lcrf, and with the aid of this new operator, we study the uniqueness of meromorphic functions $f(z)$f(z) and $L_{c}^{r}\,f$Lcrf sharing two or more values in the extended complex plane. The results obtained in the paper significantly improve a number of existing results. Further, using the notion of weighted sharing of sets, we deal the same problem. We exhibit a handful number of examples to justify certain statements relevant to the content of the paper. We are also able to determine the form of the function that coincides with its reduced linear c-shift operator. At the end of the paper, we pose an open question for future research.

Let f be a transcendental meromorphic function of finite order and c be a nonzero complex number. Define $\Delta _{c}f=f(z+c)-f(z)$ Δ c f = f ( z + c ) − f ( z ) . The authors investigate the existence on the fixed points of $\Delta _{c}f$ Δ c f . The results obtained in this paper may be viewed as discrete analogues on the existing theorem on the fixed points of $f'$ f ′ . The existing theorem on the fixed points of $\Delta _{c}f$ Δ c f generalizes the relevant results obtained by (Chen in Ann. Pol. Math. 109(2):153–163, 2013; Zhang and Chen in Acta Math. Sin. New Ser. 32(10):1189–1202, 2016; Cui and Yang in Acta Math. Sci. 33B(3):773–780, 2013) et al.

This paper is concerned with description of the existence and the forms of entire solutions of several second-order partial differential-difference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.

Let c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

In this paper, we prove the following result: Let f be a nonconstant meromorphic function of finite order, p be a nonconstant polynomial, and c be a nonzero constant. If f, $\Delta _{c}f$ , and $\Delta_{c}^{n}f$ ( $n\ge 2$ ) share ∞ and p CM, then $f\equiv \Delta_{c}f$ . Our result provides a difference analogue of the result of Chang and Fang in 2004 (Complex Var. Theory Appl. 49(12):871–895, 2004).

For the nonlinear difference equations of the form $$ w(z + 1)w(z - 1) = h(z)w^{m}(z), $$ where $h(z)$ is a nonzero rational function and $m = \pm 2, \pm 1,0$ , we show that its transcendental meromorphic solution is mainly determined by its zeros, 1-value points and poles except for some special cases. Examples for the sharpness of these results are given.

This paper mainly concerns the uniqueness of meromorphic solutions of first order linear difference equations of the form * $$ R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z), $$ where $R_{1}(z)\not \equiv 0$ , $R_{2}(z)$ , $R_{3}(z)$ are rational functions. Our results indicate that the finite order transcendental meromorphic solution of equation (*) is mainly determined by its zeros and poles except for some special cases. Examples for the sharpness of our results are also given.

In this paper, we study the admissible meromorphic solutions to the algebraic differential equation $f^{n} f' + P_{n-1}( f ) = u\mathrm {e}^{v}$ in an angular domain instead of the whole complex plane, where $P_{n-1}(f)$ is a differential polynomial in f of degree $\leq n-1$ with small function coefficients, u is a non-vanishing small function of f and v is an entire function. Herein, mainly, we are able to show that the equation does not admit any meromorphic solution f under some conditions unless $P_{n-1}(f)\equiv0$ . Using this result, we are able to extend or generalize a well-known result of Hayman.

In this paper, we study the uniqueness problems of meromorphic functions and their difference operators. Our main result is a difference analogue of a result of Jank–Mues–Volkmann, which is concerned with the uniqueness of an entire function sharing one finite value with its derivatives. Some recent papers studied the case of entire functions of finite order sharing a periodic small function to f. We consider the case of meromorphic functions of finite order sharing a polynomial, which is a more popular case. Examples are provided for our results.

We prove that if $g(z)$ is a finite-order transcendental meromorphic solution of $$\bigl(\triangle_{c} g(z)\bigr)^{2}=A(z)g(z)g(z+c)+B(z), $$ where $A(z)$ and $B(z)$ are polynomials such that $\deg A(z)>0$ , then $$1 \leq\rho(g)=\max\biggl\{ \lambda(g), \lambda\biggl(\frac{1}{g}\biggr) \biggr\} . $$

In this paper, relying on Nevanlinna theory of the value distribution of meromorphic functions, we mainly study meromorphic solutions of certain types of q-difference differential equations, obtain estimates of the growth order of their meromorphic solutions, and give a number of examples to show what our results are the best possible in certain senses. Improvements and extensions of some results in the literature are presented.

Consider the difference equation $$\biggl[\frac{\Delta^{3}f(z)}{\Delta f(z)}-\frac{3}{2} \biggl(\frac {\Delta^{2}f(z)}{\Delta f(z)} \biggr)^{2} \biggr]^{k} =\frac{P(z,f(z))}{Q(z,f(z))}, $$ where $P(z,f)$ and $Q(z,f)$ are prime polynomials in $f(z)$ with $\deg_{f}P=p, \deg_{f}Q=q$ , and $d=\max\{p,q\}>0$ . We give the supremum of d, an estimation of the sum of Nevanlinna exceptional values of meromorphic solution $f(z)$ of the equation, and study the value distributions of their difference $\Delta f(z)$ and divided difference $\frac{\Delta f(z)}{f(z)}$ .

Using Nevanlinna theory of the value distribution of meromorphic functions, the growth of entire solutions and the form of transcendental meromorphic solutions of some types of systems of higher-order complex difference equations are investigated. Some new results are obtained. We also investigate the problem of the existence of solutions of complex q-difference equations, and we obtain some new results, which are different from analogue differential equations. Improvements and extensions of some results in the literature are presented. Some examples show that our results are, in a sense, the best possible.

In this paper, we mainly investigate some properties of the transcendental meromorphic solution $f(z)$ for the difference Riccati equation $f(z+1)=\frac{p(z)f(z)+q(z)}{f(z)+s(z)}$ . We obtain some estimates of the exponents of the convergence of the zeros and poles of $f(z)$ and the difference $\Delta f(z)=f(z+1)-f(z)$ .

The main purpose of this paper is to give the Malmquist type result of the meromorphic solutions of a system of complex difference equations of the following form: $$\left \{ \textstyle\begin{array}{l} \sum_{\lambda_{1} \in I_{1}, \mu_{1}\in J_{1}}\alpha_{\lambda_{1}, \mu_{1}}(z) (\prod_{\nu=1}^{n}f(z+c_{\nu})^{l_{\lambda_{1}, \nu}}\prod_{\nu=1}^{n}g(z+c_{\nu})^{m_{\mu_{1}, \nu}} ) = \frac{\sum_{i=0}^{p}a_{i}(z)g(z)^{i} }{\sum_{j=0}^{q}b_{j}(z)g(z)^{j}}, \\ \sum_{\lambda_{2} \in I_{2}, \mu_{2}\in J_{2}}\beta_{\lambda_{2}, \mu_{2}}(z) (\prod_{\nu =1}^{n}f(z+c_{\nu})^{l_{\lambda_{2}, \nu}}\prod_{\nu=1}^{n}g(z+c_{\nu})^{m_{\mu_{2}, \nu}} ) = \frac{\sum_{k=0}^{s}d_{k}(z)f(z)^{k} }{\sum_{l=0}^{t}e_{l}(z)f(z)^{l}}, \end{array} \right . $$ where $c_{1}, c_{2}, \ldots, c_{n}$ are distinct, nonzero complex numbers, the coefficients $\alpha_{\lambda_{1}, \mu_{1}}(z)$ ( $\lambda _{1} \in I_{1}$ , $\mu_{1}\in J_{1}$ ), $\beta_{\lambda_{2}, \mu_{2}}(z)$ ( $\lambda_{2} \in I_{2}$ , $\mu_{2}\in J_{2}$ ), $a_{i}(z)$ ( $i=0,1,\ldots, p$ ), $b_{j}(z)$ ( $j=0,1,\ldots, q$ ), $d_{k}(z)$ ( $k=0,1,\ldots, s$ ), and $e_{l}(z)$ ( $l=0,1,\ldots, t$ ) are small functions relative to $f(z)$ and $g(z)$ , $I_{i} = \{\lambda _{i}=(l_{\lambda_{i}, 1}, l_{\lambda_{i}, 2}, \ldots, l_{\lambda_{i},n})| l_{\lambda_{i}, \nu}\in{N} \cup\{0\},\nu= 1,2,\ldots,n\}$ ( $i=1,2$ ) and $J_{j} = \{\mu_{j}=(m_{\mu_{j}, 1}, m_{\mu_{j}, 2}, \ldots, m_{\mu_{j},n})| m_{\mu_{j}, \nu}\in{N} \cup\{0\},\nu=1,2, \ldots,n\}$ ( $j=1, 2$ ) are finite index sets. The growth of meromorphic solutions of a related system of complex functional equations is also investigated.

In this paper, we investigate the existence and growth of solutions of the q-difference equation $\prod_{i=1}^{n}f(q_{i}z)=R(z,f(z))$ , where $R(z,f(z))$ is an irreducible rational function in $f(z)$ . We also give an estimation of the growth of transcendental meromorphic solutions of the equation $\prod_{i=1}^{n}f(q_{i}z)=f(z)^{m}$ .

In this paper, we investigate the growth of meromorphic solutions of some linear difference equations. We obtain some new results on the growth of meromorphic solutions when most coefficients in such equations are meromorphic functions, which are supplements of previous results due to Li and Chen (Adv. Differ. Equ. 2012:203, 2012) and Liu and Mao (Adv. Differ. Equ. 2013:133, 2013).

In this paper, we investigate Borel exceptional values of meromorphic solutions of Painleve III difference equations. In particular, let w be a transcendental meromorphic solution of with finite order, where η (≠0), λ ( ) are constants. If a, b are two Borel exceptional values of w, then and . MSC:30D35, 39A10.

For a transcendental entire function in the complex plane, we study its divided differences . We partially prove a conjecture posed by Bergweiler and Langley under the additional condition that the lower order of is smaller than . Furthermore, we prove that if zero is a deficient value of , then , where . MSC:30D35.

In this paper, we investigate the growth of transcendental meromorphic solutions of some types of systems of complex functional equations and obtain the lower bounds for Nevanlinna lower order for meromorphic solutions of such equations. Our results are improvement of the previous theorems given by Gao, Zheng and Chen. Some examples are also given to illustrate our results.

In this paper, we investigate the properties of meromorphic solutions of Painleve III difference equations. In particular, we study the existence of Borel exceptional value, the exponent of convergence of zeros, poles and fixed points of a transcendental meromorphic solution.

In this paper, we study zeros of some difference polynomials in f (z) and their shifts, where f (z) is a finite order meromorphic function having deficient value ∞. These results improve previous findings.

This paper is devoted to studying the growth of meromorphic solutions of some linear difference equations. We obtain some results on the growth of meromorphic solutions when most coefficients in such equations have the same order, which are supplements of previous results due to Chiang and Feng, and Laine and Yang. Some examples are given to show the sharpness of our results.

In this paper, we consider the value distribution of meromorphic solutions for linear difference equations with meromorphic coefficients.

In this paper, we investigate meromorphic solutions of linear difference equations and prove a number of results. We give estimates for the growth of meromorphic solutions under some special cases and provide some examples to show that the answer to a question of Laine and Yang is not always positive. The zeros, poles and fixed points of finite order solutions are also studied.

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

The main purpose of this paper is to present the properties of the meromorphic solutions of complex difference equations of the form , where is a collection of all subsets of , are distinct, nonzero complex numbers, is a transcendental meromorphic function, 's are small functions relative to , and is a rational function in with coefficients which are small functions relative to .

The main purpose of this paper is to present some properties of the meromorphic solutions of complex difference equation of the form , where and are two finite index sets, are distinct, nonzero complex numbers, and are small functions relative to is a rational function in with coefficients which are small functions of . We also consider related complex functional equations in the paper.

In this paper, by using Nevanlinna value distribution theory, we consider a certain type of difference equation, which originates with the difference Painleve I equation, $f(z+1)+f(z-1)=\frac{A(z)}{f(z)}+C(z)$ , where $A(z)$ , $C(z)$ are small meromorphic functions relative to $f(z)$ , and we obtain the existence and the forms of rational solutions. We also discuss the properties of the Borel exceptional value, zeros, poles, and fixed points of finite order transcendental meromorphic solutions.

In this paper, we investigate the growth and the exponent of convergence of the sequence of φ-points of meromorphic solutions of the linear differential equations A k ( z ) f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 $$A_{k}(z)f^{(k)}+A_{k-1}(z)f^{(k-1)}+ \cdots+A_{1}(z)f'+A_{0}(z)f=0 $$ and A k ( z ) f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = F ( z ) , $$A_{k}(z)f^{(k)}+A_{k-1}(z)f^{(k-1)}+ \cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$ with entire coefficients A j ( z ) $A_{j}(z)$ , j = 0 , 1 , … , k $j=0,1,\ldots,k$ and F ( z ) $F(z)$ , where k ≥ 2 $k\geq2$ , A 0 ( z ) A k ( z ) ≢ 0 $A_{0}(z)A_{k}(z)\not\equiv0$ , φ ( z ) $\varphi(z)$ is a meromorphic function of finite order, and there is only one dominant coefficient A k ( z ) $A_{k}(z)$ of the maximal order, which is also a Fabry gap series.