Value sharing results for shifts of meromorphic functions

In this paper, we obtain difference analogues of the second main theorem for meromorphic functions in several complex variables from which difference analogues of Picard-type theorems are also obtained. Our results are improvements or extensions of some recent results of papers [Proc. Royal Soc. Edinburgh Section A Math. 137, 457–474 (2007); Comput. Meth. Funct. Theor. 12, No. 1, 343–361 (2012)]. The method we used is very different from theirs.

Difference analogues of the second main theorem for meromorphic functions in several complex variables

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

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Results on meromorphic functions sharing three values with their difference operators

We prove a uniqueness theorem of nonconstant meromorphic functions sharing three distinct values IM and a fourth value CM with their shifts, and prove a uniqueness theorem of nonconstant entire functions sharing two distinct small functions IM with their shifts, which respectively improve Corollary 3.3(a) and Corollary 2.2(a) from [12], where the meromorphic functions and the entire functions are of hyper order less than 1. An example is provided to show that the above results are the best possible. We also prove two uniqueness theorems of nonconstant meromorphic functions sharing four distinct values with their difference operators.

http://koreascience.or.kr:80/article/JAKO201624238395670.pdf

Meromorphic functions sharing four values with their difference operators or shifts

In this article, we investigate possible relationships between Picard values of entire functions f(z) and their forward differences Δ n f(z). We obtain difference analogues of results of Hayman-Milloux and Tumura.

Relationships between entire functions and their forward differences

/pdf/value-distribution-of-difference-polynomials-23u61qpmeg.pdf

Zhihao Tang

Papers

Value Distribution of Difference Polynomials