English

In this paper, we shall study the unicity of meromorphic functions definedover non-Archimedean fields of characteristic zero such that their valence func-tions of poles grow slower than their characteristic functions. If f is such afunction, and f and a linear differential polynomial P(f) of f, whose coeffi-cients are meromorphic functions growing slower than f, share one finite valuea CM, and share another finite value b (6= a) IM, then P(f) = f. 1 Introduction. In 1929, R. Nevanlinna studied the unicity of meromorphic functions in C. Thefive value theorem due to R. Nevanlinna states that if two non-constant meromor-phic functions f and gin C share five distinct complex numbers a j IM (ignoringmultiplicity), which meansf −1 (a j ) = g −1 (a j ), j= 1,2,...,5in the sense of sets, then it follows that f = g. The four value theorem of R.Nevanlinna states that if two non-constant meromorphic functions f and gin Cshare four distinct complex numbers a j CM (counting multiplicity), which meansf

Mues–Reinders proved that if a non-constant meromorphic function f shares three distinct finite values IM with its Kth (2≤ k≤ 50) linear differential polynomial L(f), then f≡L(f). Frank and Hua proved the same result under a weak condition k≥2. In the present paper, we will prove this result in the case that k = 1.

Using the results of S. S. Bhoosnurmath, we mainly study the uniqueness of entire and meromorphic functions that share small functions with their homogeneous and linear differential polynomials. In this paper, we obtain significant improvements and generalizations of the results of H. X. Yi.

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by \begin{equation*} L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0. \end{equation*}In this paper, we solve an open problem posed by Li [J. Math. Anal. Appl. 310 (2005) 412-423] in connection with the problem of sharing a set by entire functions f and their linear differential polynomials $ L_k[f] $. Furthermore, we study the Fermat-type functional equations of the form $ f^n+g^n=1 $ to find the meromorphic solutions (f, g) which enable us to answer the question of Li completely. This settles the long-standing open problem of Li.

The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study a transcendental entire function f (z) that shares a non-zero polynomial a (z) with f′(z), together with its linear differential polynomials of the form: L[f] = a2(z)f″(z) + a3 (z)f′′′(z) + … + am (z)f(m) (z) (am (z) $\not\equiv$ 0), where the coefficients ak (z) (k = 2, 3, ..., m) are rational functions.

Let f be a nonconstant meromorphic function of lower order µ (f) > 1/2 in ℂ, and let aj (j = 1, 2, 3) be three distinct finite complex numbers. We show that there exists an angular domain D = {z: α ≤ arg z ≤ β}, where 0 < β − α ≤ 2π, such that if f share aj (j = 1, 2, 3) CM with its k-th linear differential polynomial L[f] in D, then f = L[f]. This generalizes the corresponding results from Frank and Schwick, Zheng and Li-Liu-Yi.

In this paper, we investigate meromorphic functions that share a small function with one of its linear differential polynomials and prove several theorems which generalize and improve the main results given by J. L. Zhang and L. Z. Yang. Some examples are provided to show that the results in this paper are sharp.

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).

On the uniqueness problems of meromorphic functions and their linear differential polynomials

In [7], Langley proved a result concerning the zeros of pairs of (possibly non-homogeneous) linear differential polynomials in a meromorphic function. We generalise this result by relaxing Langley’s assumption on the frequency of zeros (counting multiplicity), and further prove some results based on restricting the order of the differential operators.

Let f be a non-constant meromorphic function and {a=a(z)} ( {\not\equiv 0,\infty} ) a small function of f. Here, we obtain results similar to the results due to Indrajit Lahiri and Bipul Pal [Uniqueness of meromorphic functions with their homogeneous and linear differential polynomials sharing a small function, Bull. Korean Math. Soc. 54 2017, 3, 825–838] for a more general differential polynomial by introducing the concept of weighted sharing.

In the paper we prove a uniqueness theorem for meromorphic functions that share a function of slower growth with linear differential polynomials. Our result is closely related to a conjecture of Bruck.

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].

We prove a normality criterion for a family of meromorphic functions having multiple zeros which involves sharing of a non-zero value by the product of functions and their linear differential polynomials.

The paper determines all meromorphic functions f in C such that f and F have finitely many zeros, where F = f (k) + ak−1f(k−1)+ ... + a0f with k ≥ 3 and the aj rational functions. MSC 2010:30D35.