Research On Value Distribution Of Difference Polynomials

The value distribution theory founded by Rolf Nevanlinna in the1920:s. Usually, we called Nevanlinna theory in honor of him. Nevanlinna theory can be seen the most important achievements in the preceding century to understand the properties of meromorphic functions. This theory is composed of two main theorems. which are called Nevanlinna’s first and second main theorems that had been significant breakthroughs in the development of the classic function the-ory, since the Nevanlinna’s second theorem generalizes and extends Picard’s first theorem greatly, and hence it denoted the beginning of the theory of meromor-phic functions. Since then, Nevanlinna theory and its extensive has numerous applications in some fields of mathematics, for example, uniquess theory, several complex variables, complex differential, complex difference equations and so on Meanwhile, in view of the beauty of Nevanlinna theory, many outstanding math-ematicians founded and developed the value distribution theory of meromorphic mappings over certain complex manifolds.The basis of complex difference equation originated in early1920s. Batchelder [2], Norlund [41], and Whittaker [44] made a great contribution in this field. Later on, by applying Nevanlinna’s value distribution theory of meromorphic functions, Shimomura [42] and Yanagihara [45,46,47] investigate the existence of finite or-der transcendental meromorphic solutions of several kinds nonlinear difference equations. Recently, the difference analogue of the lemma on the logarithmic derivative have been obtained by Halburd-Korhonen [18]and Chiang-Feng [11], independently. Ishizaki and Yanagihara [30] deveploped a version of Wiman-Valiron theory for difference equations of entire functions of small growth. Also Chiang and Feng [12] has a difference version of Wiman-Valiron. In this dissertation. by applying Nevanlinna’s value distribution theory of meromorphic functions, we investigated some properties of some difference func-tions. Nevanlinna’s theory Plays an important role. This dissertation is struc-tured as follows.In Chapter1, we recall some essential theories as background, we also intro-duce some notions which are always in our studies.In Chapter2. Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)+…+f(z+ck)-kf(z) and gk(z)=f(z+c1)f(z+c2)…f(z+ck)-fk(z). A number of results are obtained concerning the exponents of con-vergence of zeros of differences g(z), gk(z), In Chapter3, Let f be a transcendental meromorphic function and gk(z)=f(z+c1)+f(z+c2)+…+f(z+ck)-kf(z) and A number of results are obtained concerning zeros and fixed points of the differ-ence gk(z) and the divided difference G(z).In Chapter4. In this article, we investigate some properties of some difference polynomials. The results in this article improve some theorems of Liu and Laine. Several examples are provided to show that our results are best possible.In Chapter5, In this article, we investigate the uniqueness problems of dif-ferences of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Yi. An example is given to show the results in this paper are best possible.In the finial Chapter6. In this article, we investigate the growth and value distribution of meromorphic solutions of a first order difference equation with small coefficients in the complex plane.