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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 Δ η n f share a 1 , a 2 , a 3 CM, where n is a positive integer, η ≠ 0 is a finite complex value, and a 1 , a 2 , a 3 are three distinct finite complex values, then f (z) = Δ η n f (z) for each z ∈ C. The main results in this paper improve and extend many known results concerning a conjecture posed by Chen and Yi in 2013. [ABSTRACT FROM AUTHOR]

By Cartan's version of Nevanlinna's theory, we prove the following result: let m and n be two positive integers satisfying n ≥ 2 + m , let p ≢ 0 be a polynomial, let η ≠ 0 be a finite complex number, let ω 1 , ω 2 , ... , ω m be m distinct finite nonzero complex numbers, and let H j be either exponential polynomials of degree less than q, or an ordinary polynomial in z for 0 ≤ j ≤ m , such that H j ≢ 0 for 1 ≤ j ≤ m. Suppose that f ≢ ∞ is a meromorphic solution of the difference equation: 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 ρ 2 (f) < 1. Then, f reduces to a transcendental entire function, such that either n = m + 2 with H 0 ≢ 0 and λ (f) = ρ (f) = q , or m = 2 , H 0 = 0 and: f (z) = H 1 (z - η) e ω 1 (z - η) q p (z - η) with 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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, ), Frank and Schick (Results Math 22:679-684, ), Bernstein et al. (Forum Math 8:379-396, ) and improve Theorem 3 in Zheng (Can Math Bull 47:152-160, ). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

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 [] by Liu and Yang and deal with Question 1 in [], where the entire functions are of finite order. Moreover, we repair certain statements in [] by Li et al., which in turn had depended on questionable assertions of Lemma 2.6 in []. Examples are provided to show that the results in this paper are best possible. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]