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In this paper, we prove that if a nonconstant finite order meromorphic function f and its n-th order difference operator $$\Delta ^n_{\eta }f$$ share $$a_1,$$ $$a_2,$$ $$a_3$$ CM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, and $$a_1,$$ $$a_2,$$ $$a_3$$ are three distinct finite complex values, then $$f(z)=\Delta ^n_{\eta }f(z)$$ for each $$z\in \mathbb {C}.$$ The main results in this paper improve and extend many known results concerning a conjecture posed by Chen and Yi in 2013.

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and $$f(z+\eta )$$ share a, b, c IM, where $$\eta $$ is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then $$f(z)=f(z+\eta )$$ for all $$z\in \mathbb {C}$$ . We prove that if a non-constant finite order entire function f and its n-th difference operator $$\Delta ^n_{\eta }f$$ share $$a_1$$ , $$a_2$$ , $$a_3$$ IM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, while $$a_1$$ , $$a_2$$ , $$a_3$$ are three distinct finite complex values, then $$f=\Delta ^n_{\eta }f$$ . The main results in this paper also improve Theorems 1.1 and 1.2 from Li and Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).

We study a uniqueness question of meromorphic functions whose certain nonlinear differential polynomials share a finite nonzero value. The results in this paper extend the corresponding results from Steuding [14, p.152], Li[9] and Fang [1]. The studied question is concerning a question posed by Fang in 2009.

By Cartan’s version of Nevanlinna’s theory, we prove the following result: let m and n be two positive integers satisfying $$n\ge 2+m,$$ n ≥ 2 + m , let $$p\not \equiv 0$$ p ≢ 0 be a polynomial, let $$\eta \ne 0$$ η ≠ 0 be a finite complex number, let $$\omega _{1}, \omega _{2}, \ldots , \omega _{m}$$ ω 1 , ω 2 , … , ω m be m distinct finite nonzero complex numbers, and let $$H_{j}$$ H j be either exponential polynomials of degree less than q, or an ordinary polynomial in z for $$0\le j\le m$$ 0 ≤ j ≤ m , such that $$H_{j}\not \equiv 0$$ H j ≢ 0 for $$1\le j\le m.$$ 1 ≤ j ≤ m . Suppose that $$f\not \equiv \infty $$ f ≢ ∞ is a meromorphic solution of the difference equation: $$\begin{aligned} f^n(z)+p(z)f(z+\eta )&=H_0(z)+H_1(z)e^{\omega _{1}z^{q}}+H_2(z)e^{\omega _{2}z^{q}}\\&\quad +\cdots +H_m(z)e^{\omega _{m}z^{q}}, \end{aligned}$$ f n ( z ) + p ( z ) f ( z + η ) = H 0 ( z ) + H 1 ( z ) e ω 1 z q + H 2 ( z ) e ω 2 z q + ⋯ + H m ( z ) e ω m z q , such that the hyper-order of f satisfies $$\rho _2(f) ρ 2 ( f ) < 1 . Then, f reduces to a transcendental entire function, such that either $$n=m+2$$ n = m + 2 with $$H_0\not \equiv 0$$ H 0 ≢ 0 and $$\lambda (f)=\rho (f)=q,$$ λ ( f ) = ρ ( f ) = q , or $$m=2,$$ m = 2 , $$H_0=0$$ H 0 = 0 and: $$\begin{aligned} f(z)=\frac{H_1(z-\eta )e^{\omega _{1}(z-\eta )^{q}}}{p(z-\eta )} \end{aligned}$$ f ( z ) = H 1 ( z - η ) e ω 1 ( z - η ) q p ( z - η ) with $$\begin{aligned} H^n_1(z)=p^n(z)H_2(z+\eta )e^{\omega _2P_{q-1}(z)}\quad \text {and}\quad P_{q-1}(z)=\sum \limits _{k=1}^q\left( {\begin{array}{c}q\\ k\end{array}}\right) \eta ^kz^{q-k}. \end{aligned}$$ H 1 n ( z ) = p n ( z ) H 2 ( z + η ) e ω 2 P q - 1 ( z ) and P q - 1 ( z ) = ∑ k = 1 q q k η k z q - k . This result improves Theorems 1.1 and 1.3 from [19] by removing some assumptions of theirs. An example is provided to show that some results obtained in this paper, in a sense, are the best possible.

We study the value distribution of L-functions and uniqueness questions of meromorphic functions sharing one or two sets with L-functions. The results in this paper extend the corresponding results from [J. Steuding, Value Distribution of L-Functions, Lect. Notes Math., Vol. 1877, Springer-Verlag, Berlin, Heidelberg, 2007, p. 152] and [B.Q. Li, A result on value distribution of L-functions, Proc. Am. Math. Soc., 138(6):2071–2077, 2010]. The studied questions in this paper are related to the questions of Gross from [F. Gross, Factorization of meromorphic functions and some open problems, in Complex Analysis, Kentucky 1976. Proceedings of the Conference Held at the University of Kentucky, May 18–22, 1976, Lect. Notes Math., Vol. 599, Springer-Verlag, Berlin, Heidelberg, 1977, pp. 51–69]. Here the meromorphic functions are of finitely many poles in the complex plane.

In this paper, we study a uniqueness question of meromorphic functions concerning certain linear differential polynomials that share a nonzero finite value with the same of L-functions. The results in this paper extend the corresponding results from Li[6] and Li & Li[7].

Under the restriction of finite order, we prove two uniqueness theorems of nonconstant meromorphic functions sharing three values with their difference operators, which are counterparts of Theorem 2.1 in (6) for a finite-order meromorphic function and its shift operator. 1. Introduction and main results In this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We adopt the standard notations of the Nevan- linna theory of meromorphic functions as explained in (5), (10) and (16). It will be convenient to let E denote any set of positive real numbers of finite lin- ear measure, not necessarily the same at each occurrence. For a nonconstant meromorphic function h, we denote by T(r,h) the Nevanlinna characteristic of h and by S(r,h) any quantity satisfying S(r,h) = o(T(r,h)), as r → ∞,r 6∈E. Let f and g be two nonconstant meromorphic functions, and let a be a value in the extended plane. We say that f and g share the value a CM, provided that f and g have the same a-points with the same multiplicities. We say that f and g share the value a IM, provided that f and g have the same a-points ignoring multiplicities (cf. (16)). Throughout this paper, we denote by �(f) the order of f (cf. (5), (10) and (16)). We also need the following two definitions: Definition 1.1 ((15)). Let f be a nonconstant meromorphic function. We define difference operators of f as

Let f be a transcendental meromorphic function of nite order in the plane such that f (m) has nitely many zeros for some positive integer m ≥ 2: Suppose that f (k) and f share a CM, where k ≥ 1 is a positive integer, a 0 is a nite complex value. Then f is an entire function such that f (k) − a = c(f − a); where c 0 is a nonzero constant. The results in this paper are concerning a conjecture of Bruck (5). An example is provided to show that the results in this paper, in a sense, are the best possible.

We study the uniqueness question of transcendental meromorphic functions sharing three distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane, and study the uniqueness question of transcendental entire functions sharing two distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane. The results in this paper improve the corresponding results from Frank and Weissenborn (Complex Var 7:33–43, 1986), Frank and Schick (Results Math 22:679–684, 1992), Bernstein et al. (Forum Math 8:379–396, 1996) and improve Theorem 3 in Zheng (Can Math Bull 47:152–160, 2004).

In this paper, we study the uniqueness problem of entire functions sharing polynomials IM with their first derivative. As an application, we generalize Bruck’s conjecture from sharing value CM to sharing polynomial IM for a class of functions. In fact, we prove a result as follows: Let $${a({\not\equiv} 0)}$$ be a polynomial and $${n \geq 2}$$ be an integer, let f be a transcendental entire function, and let $${F = f^n}$$ . If F and F′ share a IM, then $${f(z) = Ae^{z/n},}$$ where A is a nonzero constant. It extends some previous related theorems.

This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference polynomials of these functions have the same fixed points or share a nonzero value. This extends the research work of Qi, Yang and Liu, where they used the finite ordered meromorphic mappings.

In this paper, we study a uniqueness question of meromorphic functions of certain differential polynomials that share a nonzero finite value or have the same fixed points with the same of L-functions. The results in this paper extend the corresponding results from Steuding [12, p. 152], Li [7], Fang [1] and Yang-Hua [14].

We study the uniqueness question of transcendental meromorphic functions that share four distinct finite values a1, a2, a3, a4, where all the a j -points of the transcendental meromorphic functions are real numbers for 1 ≤ j ≤ 4. The results in this paper improve and extend the corresponding results from Czubiak–Gundersen (Proc Am Math Soc 82:393–397, 1981) and Yi (Proc Am Math Soc 124:585–590, 1996). An example is provided to show that the results in this paper, in a sense, are the best possible.

We use Zalcman’s lemma to study a uniqueness question for meromorphic functions where certain associated nonlinear differential polynomials share a nonzero value. The results in this paper extend Theorem 1 in Yang and Hua [‘Uniqueness and value-sharing of meromorphic functions’, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406] and Theorem 1 in Fang [‘Uniqueness and value sharing of entire functions’, Comput. Math. Appl. 44 (2002), 823–831]. Our reasoning in this paper also corrects a defect in the reasoning in the proof of Theorem 4 in Bhoosnurmath and Dyavanal [‘Uniqueness and value sharing of meromorphic functions’, Comput. Math. Appl. 53 (2007), 1191–1205].

Using Zalcman’s lemma, we study a uniqueness question of meromorphic functions, where the meromorphic functions and their certain non-linear differential polynomials share a non-zero value with their shifts. The results in this paper extends Theorem 1 in Yang and Hua (Ann Acad Sci Fenn Math 22:395–406, 1997) and Theorem 1 in Fang (Comput Math Appl 44:823–831, 2002). Our reasoning in this paper also corrects the proof of Theorem 4 in Bhoosnurmath and Dyavanal (Comput Math Appl 53:1191–1205, 2007).

We study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators, and deal with a question posed by Liu and Yang. The results in this paper extend the corresponding results obtained by Liu-Yang and by Liu-Laine respectively. Examples are provided to show that the results in this paper, in a sense, are the best possible.

In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.

We study the uniqueness question of transcendental entire functions sharing one finite nonzero value with their derivatives in some angular domains instead of the whole complex plane. The results in the present paper improve and extend the corresponding results from Chang-Fang~[2] and extend Theorem 3 from Zheng~[15]. An example is provided to show that the results in this paper, in a sense, are best possible.

In this paper, we study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators. The results in this paper extend Theorem 1.1 in [19] by Liu and Yang and deal with Question 1 in [19], where the entire functions are of finite order. Moreover, we repair certain statements in [21] by Li et al., which in turn had depended on questionable assertions of Lemma 2.6 in [20]. Examples are provided to show that the results in this paper are best possible.

For a meromorphic function f in the complex plane, we prove that if f is a finite order transcendental entire function which has a finite Borel exceptional value a, if \({f(z+\eta)\not\equiv f(z)}\) for some \({\eta\in \mathbb{C}}\), and if f(z + η) − f(z) and f(z) share the value a CM, then $$ a=0 \quad {\rm and} \quad \frac{f(z+\eta)-f(z)}{f(z)}=A, $$ where A is a nonzero constant. We also consider problems on sharing values of meromorphic functions and their differences when their orders are not an integer or infinite.

In this paper, we get two uniqueness theorems of meromorphic functions whose certain nonlinear differential polynomials share a polynomial. The results in this paper extend the corresponding results given by Fang (2002) in [7]. Our reasoning in this paper will correct a defective reasoning in the proof of Theorem 4 in Bhoosnurmath and Dyavanal (2007) [8]. An example is provided to show that some conditions of the main results in this paper are necessary.

In this paper, we deal with a uniqueness question of entire functions sharing a nonzero complex number with their difference operators. The results in this paper improve Theorem 1.1 in Liu and Yang (Arch. Math. 92 (2009), 270–278) and deal with Question 1 in Liu and Yang (2009), where the entire functions are of finite orders.

In this paper, we prove some uniqueness theorems of entire functions and their derivatives that share an entire of function whose order is less than their order. The results in this paper improve those given by G. G. Gundersen and L. Z. Yang, J. P. Wang, J. M. Chang, and Y. Z. Zhu, X. M. Li and H. X. Yi, and others. Some examples are provided to show that the results in this paper are the best possible.

The purpose of this article is to study the uniqueness problem for meromorphic mappings from $\mathbb{C}^{n}$ into the complex projective space $\mathbb{P}^{N}(\mathbb{C}).$ By making using of the method of dealing with multiple values due to L. Yang and the technique of Dethloff-Quang-Tan respectively, we obtain two general uniqueness theorems which improve and extend some known results of meromorphic mappings sharing hyperplanes in general position., Comment: 10 pages

In the paper, we investigate the uniqueness problem of a transcendental meromorphic function f that shares a rational function with its first derivative f ′ , together with its linear differential polynomials of constant coefficients.

In the paper, we study the uniqueness and the shared fixed-points of meromorphic functions and prove two main theorems which improve the results of Fang and Fang and Qiu.

We prove a theorem on the growth of nonconstant solutions of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its linear differential polynomial share a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.

In this paper, we estimate the size of ρ n 's in the famous L. Zalcman's lemma. With it, we obtain some uniqueness theorems for meromorphic functions f and f ′ when they share two transcendental meromorphic functions.

The main purpose of this paper is to investigate the uniqueness of transcendental meromorphic functions that share four values in one angular domain which is an unbounded subset of the whole complex plane. From one of our main results, a question of J.H. Zheng [J.H. Zheng, On uniqueness of meromorphic functions with shared values in one angular domain, Complex Var. Elliptic Equ. 48 (9) (2003) 777–785] is completely answered. Furthermore, we give an example to explain the necessity of the condition lim r → ∞ S α , β ( r , f ) log ( r T ( r , f ) ) = ∞ in our results.

In this paper, we investigate the growth and fixed points of meromorphic solutions of higher order linear dierential equations with meromorphic coecients and their derivatives. Because of the restriction of dierential equations, we obtain that the properties of fixed points of meromorphic solutions of higher order linear dierential equations with meromorphic coecients and their derivatives are more interesting than that of general transcendental meromorphic functions. Our results extend the previous results due to M. Frei, M. Ozawa, G. Gundersen, and J. K. Langley and Z. Chen and K. Shon.

Using the notion of weighted sharing of values we prove two uniqueness theorems which improve the results proved by T.C. Alzahary [T.C. Alzahary, Weighted sharing three values and Brosch’s theorem. J. Math. Anal. Appl. 323 (2006) 8–25; T.C. Alzahary, On a result of Ueda concerning unicity of meromorphic functions. Kodai. Math. J. 30 (2007) 140–146], under a weaker hypothesis.

In this paper, we investigate the complex oscillation theory of the second order linear differential equation f ″ + A (z)f = 0, where the coefficient A (z) is an analytic function in the unit disc Δ = {z: |z | < 1} (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, we will prove some uniqueness theorems of meromorphic functions whose derivatives share four distinct small functions. The results in this paper improve those given by R. Nevanlinna, L. Yang, G.D. Qiu, and other authors. An example is provided to show that the results in this paper are best possible.

In this paper, we discuss the higher order linear differential equation f ( k ) + A k − 1 f ( k − 1 ) + ⋯ + A 1 f ′ + A 0 f = 0 on an angle. We apply Nevanlinna sectorial theory to estimate the growth of solutions of the equation restricted in the angle and prove the existence of asymptotic value of the derivative of solutions of the equation in the angle. By using these results we can improve some known results in the plane.