Research On Value Distribution And Value Sharing Problem Of Difference Polynomials And Differential Polynomials

After Nevanlinna's [47] profound work,the value distribution theory has acquired, in some way, its complete form. This theory, together with subsequent contributions, occupies one of the central places in complex analysis. In spite of a certain completeness of the value distribution theory, the study of even classical problems has not been brought to an end. On the contrary, it becomes more and more extensive. Extensive research is devoted to its connections and applications in other areas of mathematics such as several complex variables, complex differential, difference and functional equations etc.The uniqueness theory of meromorphic functions mainly studies conditions under which there exists essentially only one function satisfying the conditions. The classical results in the uniqueness theory of meromorphic functions are the five-point, resp. four-point, theorems due to Nevanlinna. During the last two decades, uniqueness theory of meromorphic functions has developed to various directions, such as uniqueness studies of a meromorphic function and its derivative, see [60].As a recent analogue, Heittokangas [26,27] et al. started to consider the uniqueness of a meromorphic function sharing values with its shift. The background for these considerations lies in the recent difference counterparts of Nevanlinna theory. The key result here is the difference analogue of the lemma on the logarithmic derivative obtained by Halburd-Korhonen [20,21] and Chiang-Feng [5], independently.In this dissertation, we mainly study value sharing for difference operators of mero- morphic functions, which can be seen as an application of Nevanlinna's value distribution theory. This dissertation is structured as follows:In Chapter 1, we recall some essential theories as background, we also introduce some notions which are always in our studies.In order to understand the main ideas in Chapters 3-6, we give some preceding results in Chapter 2.In Chapter 3, we consider the existence of transcendental entire solutions of certain type of a non-linear difference equation. Indeed, the non-linear difference equation we study here is related to the equation in [33,61]. The difference is that the degree of the differential-difference polynomial is equal to the degree of f in this chapter. We get the following result:Theorem 0.1. Let a, c, A be non-zero complex constants, n and m be integers satisfying n≥m>0, and let P(z), Q(z) be polynomials. If n≥2, then the difference equation f(z)n+m+λf(z)nf(z+c)m=P(z)eQ(z)+a (0.0.2) has no transcendental entire solutions of finite order.As an application of Theorem 0.1, we investigate the value distribution of difference polynomials of entire functions.Theorem 0.2. Let f be a transcendental entire function of finite order, c be a non-zero constant, n and m be integers satisfying n≧m> 0, and letλ,μbe two complex numbers such that |λ|+|μ|≠0. If n≥2, then either f(z)n(λf(z+c)m+μf(z)m) assumes every non-zero value a∈C infinitely often or f(z)≡e(?)zg(z), where t=((?)) and g(z) is a periodic function with period c.Corresponding to Theorem 0.2, we consider the value distribution of f(z)n+μf(z+ c)m, where m≠n. Moreover, we also give a partial answer as to what may happen if n=m in Theorem 0.4.Theorem 0.3. Let f be a transcendental entire function of finite order,μand c be non-zero constants, and let a(z) be a non-zero small function to f. Suppose that n, m are positive integers such that n>m+1 (or m>n+1). Then the difference polynomial f(z)n+μf(z+c)m - a(z) has infinitely many zeros.Theorem 0.4. Let f be an entire function of order 1≤ρ(f)<∞, and suppose that f has infinitely many zeros with the exponent of convergence of zerosλ(f)<1. Let n be a positive integer, and letμ, a and c be non-zero complex constants such that f(z)n+μf(z+c)n(?)0. Then the difference polynomial f(z)n+μf(z+c)n-a has infinitely many zeros.Concerning results in Chapter 3, we investigate the uniqueness problems for differ-ence polynomials of entire functions in Chapter 4. The main results read as follows:Theorem 0.5. Let f and g be transcendental entire functions of finite order, c be a non-zero complex constant, and letn≥6 be an integer. If fnf(z+c) and gng(z+c) share 1 CM, then fg≡t1 or f≡t2g for some constants t1 and t2 that satisfy t1n+1=1 and t2n+1=1.Note that sharing the identity mapping z means sharing fixed points. In this chap-ter, we prove the following result concerning fixed points.Theorem 0.6. Let f and g be transcendental entire functions of finite order, c be a non-zero complex constant, and let n≥6 be an integer. If fnf(z+c) and gng(z+c) share z CM, then f≡tg for a constant t that satisfies tn+1=1.Very recently, Zhang[65, Theorem 6] has obtained the same result of next theorem. However, our proof is different to the one in [65].Theorem 0.7. Let f and g be transcendental entire functions of finite order, let c be a non-zero complex constant, and let n≥7 be an integer. If f(z)n(f(z)-1)f(z+c) and g(z)n(g(z)-1)g(z+c) share a CM, where a∈S(f)∩S(g)\{0}, then f(z)≡g(z).Recently, Heittokangas [26,27] et al showed that if a finite order entire function f(z) and f(z+c) share two distinct small functions a1 IM and a2 CM with period c, then f(z)=f(z+c). In Chapter 5, we show that "1CM+1IM" can be replaced by "2IM". We also consider the case that F=fn, assuming value sharing with F and F(z+c). This result also can be seen as a difference analogue to the Bruck Conjecture.In the last chapter, we return to consider the value sharing problem of differential polynomials, and we improve some previous results. Moreover, we give a new condition that make Briick Conjecture hold in this chapter.