Applications Of Nevanlinna Theory In Difference Polynomials

After a number of shorter notes in the years 1922-1925, Nevanlinna published his fundamental article [51] on the theory of meromorphic functions, later known as Nevanlinna theory.And its geometric form was given by L. Ahlfors about a decade later. Nevanlinna theory can beseen the most important achievements in the preceding century to understand the propertiesof meromorphic functions. This theory, together with subsequent contributions, now forms asignificant part of modern function theory. Moreover, Nevanlinna theory and its extensive hasnumerous applications in some fields of mathematics, for example, potential theory, complexdifferential and difference equations, several complex variables, minimal surfaces, and so on.The foundation of the theory of complex difference equations was laid by Batchelder [2],N(?)rlund [52], and Whittaker [57] in the early twentieth century. Later on, Shimomura [55]and Yanagihara [59, 60, 61] studied nonlinear complex difference equations from the viewpointof Nevanlinna theory. Meromorphic solutions of complex difference equations have become asubject of some interest recently, due to the fact that the existence of finite order solutions is agood detector of integrability of difference equations [23]. In such considerations, Nevanlinnatheory appears to be a powerful tool.Difference counterparts of Nevanlinna theory have been established very recently. Thekey result is the difference analogue of the lemma on the logarithmic derivative obtained byHalburd-Korhonen [20] and Chiang-Feng [8], independently. Halburd and Korhonen [21] alsoestablished a version of Nevanlinna theory for difference operators. Ishizaki and Yanagihara [33]developed a version of Wiman-Valiron theory for slowly growing entire solutions of differenceequations. Bergweiler and Langley [4, 38] considered the value distributions of differenceoperators of slowly growing meromorphic functions.This dissertation is devoted to investigating the value distributions of difference polynomial of meromorphic functions. Nevanlinna theory takes an important part in the dissertation.The dissertation is structured as follows:In Chapter 1, we recall the basic background of Nevanlinna theory and some notationswhich arc always used in our studies. It also includes some classical results in uniqueness theory of values sharing.In Chapter 2, we shortly recall difference analogues of the lemma on the logarithmicderivative, of the Clunie lemma, of the second main theorem and their consequences. Someimportant results about the existence and growth of solutions of difference equations anddifference analogue of Briick conjecture also can be found in this chapter.In Chapter 3, the value distributions of difference products will be introduced. We alsoobtain some results, which can be seen the difference analogues of classical results given byHayman, i.e. the value distributions of differential polynomials of the type fⁿf'. In fact, weobtained the following result.Theorem 0.1. Let f(z) be a transcendental entire function of finite order, let c be a non-zeroconstant, and let n≥2 be an integer. Then both f(z)ⁿf(z+c)-p(z) and f(z)ⁿâ–³_cf-p(z)have infinitely many zeros, where p(z)(?)0 is a polynomial in z.For more generally difference products that the case f is a transcendental meromorphicfunctions, we consider the value distributions of difference products of meromorphic functionsof the formΠ_j=1ⁿf(z+c_j)^v_j, where c_j∈C are distinct constants. We not only improveTheorem 0.1, but also obtain a quantitative version:Theorem 0.2. Let f be a transcendental meromorphic function with finite order p(f), S(z)=R(z)e^Q(z), where R(z) is a non-zero rational function, Q(z) is a polynomial that satisfiesdegQ(z)＜ρ(f), andλ(?)＜ρ(f). If (?), and at least one v_j≥2, thenΠ_j=1ⁿf(z+c_j)^v_j-S(z) has infinitely many zeros. If there exists a unique large exponent in the sense ofthenIn addition, we also investigate the value distributions of certain difference operators ofmeromorphic functions. The main idea is to find that properties of differences are somewhatsimilar to known properties of derivatives. Namely, some results are proved concerning theexistence of zeros of f^kΔ_cf-a(z),k∈N∪{0}. This expression can be viewed as a discreteanalogue of f^kf'-a(z), see Hayman [27]. Our results can be stated as follows:Theorem 0.3. Let f be a meromorphic function of order 1≤ρ(f)＜∞, and let a,c∈C\ {0}such thatâ–³_cf(?)0. Suppose that f has infinitely many zeros withλ(f)＜1. If f has finitelymany poles, thenâ–³_cf-a has infinitely many zeros. Theorem 0.4. Let f be a transcendental meromorphic function of orderρ(f)＜1, c be anon-zero constant, and let B= {b_j} be a set consisting of all poles of f such that b_j + kc (?).B(k=1,…,m) with at most finitely many exceptions. Then f(z)ⁿâ–³_cf-a has infinitelymany zeros.In this section, it also contains several examples showing that various assumptions ofresults are non-redundant.In Chapter 4, we want to introduce some results which can be related the value distributions of difference polynomials. We first recall two classical theorems due to Hayman [25,Theorem 8 & 9]. Then our results can be stated as follows:Theorem 0.5. Let f be a transcendental meromorphic function of finite orderρ(f)=ρ, notof period c,λ(?)＜ρ(f), s be rational and a be a non-zero constant. Then the differencepolynomial f(z)ⁿ+aΔ_cf-s(z) has infinitely many zeros in the complex plane, provided n≥3,resp. n≥2 if s = 0.Theorem 0.6. Let f be a transcendental meromorphic function of finite orderρ(f)=ρ, not ofperiod c, a be a non-zero complex constant. Then the difference polynomial f(z)ⁿ+aΔ_cf-s(z)has infinitely many zeros in the complex plane, provided n≥8.In the finial Chapter 5, we obtain some uniqueness type results for an entire functionf(z) that shares a common set with its shift f(z + c) or with its difference operatorâ–³_cf. Ourresults can be understood as difference counterparts to f(z) sharing a common set with itsderivatives, see [39]. One of the main results reads as follows:Theorem 0.7. Let f be a transcendental entire function of finite order, c∈C\{0}, and leta(z)∈S(f) be a non-vanishing periodic entire function with period c. If f(z) and f(z + c)share the set {a(z), -a(z)} CM, then f(z) must take one of the following conclusions:If f(z) andâ–³_cf share the set {a, -a} CM, where a∈C, then similar conclusions as thosein Theorem 0.7 can be found in the chapter as well.We consider the solutions of nonlinear difference equations using above theorem. We givethe entire solutions of finite order of nonlinear difference equation f(z)²+f(z+c)²=a(z)²,while we conclude that there does not exist a non-constant entire function of finite ordersatisfying the nonlinear difference equation f(z)²+(â–³_cf)²=a². A partial answer is given to an open problem proposed by C. C. Yang at his colloquiumtalk in the University of Joensuu. Namely, let h(z) be a polynomial, and let b∈C and n≥3.Then the equation f(z)ⁿ+ bf(z + c)- h(z) has no infinite order entire solution f satisfying(?).We also continue to relax the value sharing conditions in the result [32, Theorem 8]. Ourresult shows that f(z) is c-periodic only.