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In the paper, we use the idea of normal family to investigate the uniqueness problems of entire functions when certain types of differential-difference polynomials generated by them sharing a non-zero polynomial. Also we exhibit one example to show that the conditions of our results are the best possible.

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This paper surveys some surprising applications of a lemma char-acterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol'dberg's Theorem (a meromorphic function on C which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree d ≥ 2 is the closure of the repelling periodic points). We also discuss Bloch's Principle and provide simple solutions to some problems of Hayman connected with this principle. Over twenty years ago, on the way to a partial explication of the phenomenon known as Bloch's Principle, I proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains [68]. Over the years, the lemma has grown and, in dextrous hands, proved amazingly versatile, with applications to a wide variety of topics in function theory and related areas. With the renewed interest in normal families 1 (arising largely from the important role they play in complex dynamics), it seems sensible to survey some of the most striking of these applications to the one-variable theory, with the aim of making this technique available to as broad an audience as possible. That is the purpose of this report. One pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity. In such cases, we have made no effort to resist the temptation to write out complete proofs. Hardly anything beyond a basic knowledge of function theory is required to understand what follows, so the reader is urged to take courage and plough on through. And now we turn to our tale.

The purpose of the paper is to study uniqueness problems of certain types of differential-difference polynomials sharing a nonzero polynomial of certain degree under relaxed sharing hypotheses. We not only point out some gaps in the proof of the main results in [

Combining value distribution theory and the classical function theory to study uniqueness theorems of meromorphic functions has become an interesting and active field in recent years. Nevanlinna himself proved his famous five-value theorem and four-value theorem right after his establishment of the value distribution theory around 1924. Since then, there appeared no significant studies on sharing values of meromorphic functions or entire functions till 1970s, several mathematician started the research in this topic (see [1], [200], [196], [281] and [280]). For instances, M. Ozawa [198] studied the properties of entire functions that share two values, while L. A. Rubel and C. C. Yang [210] studied the problems on entire functions which share two values with its derivative. G. G. Gundersen [85] and E. Mues [179] weakened the conditions of R. Nevanlinna’s four-value theorem. Since 1980s, there have many papers been published on uniqueness theory and sharing values, and a comprehensive Chinese monograph [297] was appeared in 1995. In the present chapter, we introduce and summarize some of the more recent and relatively new results on value sharing and uniqueness theory on ℂ.

In this paper, we deal with and improve one of the uniqueness results on two difference products of entire functions sharing one value by considering that the functions share the value zero, counting multiplicities. The research findings also include some IM-analogues of the theorems that we obtain, i.e. the nonzero value is allowed to be shared ignoring multiplicities. Meanwhile, we investigate the situation where the difference products share a nonzero polynomial instead, by confining its degree and generalize the previous concerning results. Moreover, we show by illustrating examples and a number of remarks that our results are best possible in certain senses.

We deal with uniqueness of entire functions whose difference polynomials share a nonzero polynomial CM, which corresponds to Theorem 2 of I. Laine and C. C. Yang [Proc. Japan Acad. Ser. A 83 (2007), 148-151] and Theorem 1.2 of K. Liu and L. Z. Yang [Arch. Math. 92 (2009), 270-278]. We also deal with uniqueness of entire functions whose difference polynomials share a meromorphic function of a smaller order, improving Theorem 5 of J. L. Zhang [J. Math. Anal. Appl. 367 (2010), 401-408], where the entire functions are of finite orders.

We investigate the growth of the Nevanlinna Characteristic of f(z+\eta)
for a fixed \eta in this paper. In particular, we obtain a precise
asymptotic relation between T(r,f(z+\eta) and T(r,f), which is only true
for finite order meromorphic functions. We have also obtained the
proximity function and pointwise estimates of f(z+\eta)/f(z) which is a
version of discrete analogue of the logarithmic derivative of f(z). We
apply these results to give growth estimates of meromorphic solutions to
higher order linear difference equations. This also solves an old
problem of Whittaker concerning a first order difference equation. We
show by giving a number of examples that all these results are best
possible in certain senses. Finally, we give a direct proof of a result
by Ablowitz, Halburd and Herbst.

We continue to studying value distribution of difference polynomials of meromorphic functions. In particular, we show that extending classical theorems of Tumura–Clunie type to difference polynomials needs additional assumptions.

In this paper, we deal with the value distribution of difference products of entire functions, and present some result on two difference products of entire functions sharing one value with the same multiplicities. The research findings also include an analogue for shift of a well-known conjecture by Brück. Our theorems improve the results of I. Laine and C.C. Yang [I. Laine, C.C. Yang, Value distribution of difference polynomials, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007) 148–151], K. Liu and L.Z. Yang [K. Liu, L.Z. Yang, Value distribution of the difference operator, Arch. Math. 92 (2009) 270–278], and J. Heittokangas et al. [J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J.L. Zhang, Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009) 352–363]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.

In this paper we study the problem of meromorphic function sharing one small function with its derivative and improve the results of K.-W. Yu and I. Lahiri and answer the open questions posed by K.-W. Yu.

We shall establish the following three results in more general forms. (1) The second main theorem for small functions. Let f be a mero-morphic function on the complex plane C. Let a1; : : : ; aq be distinct mero-morphic functions on C. Assume that ai are small with respect to f; i. e., T(r; a) < o(T(r; f)) jj. Then the inequality q X (q 2 ")T (r; f)

In this paper, we investigate the uniqueness problems of difference polynomials of meromorphic functions that share a value or a fixed point. We also obtain several results concerning the shifts of meromorphic functions and the sufficient conditions for periodicity which improve some recent results in Heittokangas et al. (2009) [10] and Liu (2009) [11].

There exists a set S with three elements such that if a meromorphic function f, having at most finitely many simple poles, shares the set S CM with its derivative f′, then f′≡f.

This research is a continuation of a recent paper due to the first four authors. Shared value problems related to a meromorphic function f(z) and its shift f(z+c), where c∈C, are studied. It is shown, for instance, that if f(z) is of finite order and shares two values CM and one value IM with its shift f(z+c), then f is a periodic function with period c. The assumption on the order of f can be dropped if f shares two shifts in different directions, leading to a new way of characterizing elliptic functions. The research findings also include an analogue for shifts of a well-known conjecture by Brück concerning the value sharing of an entire function f with its derivative f′.

Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ?, then every asymptotic value of f, except at most 2? of them, is a limit point of critical values of f.
We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f'fn with n = 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions