On The Growth Of Solutions Of Linear Complex Differential Equations And Some Complex Difference Equations

Nevanlinna theory is an important content in complex analysis.In the past many years,Much works has been taken on this issue.In this dissertation,the growth of solutions of complex linear differential equations and the growth and existence of transcendental meromorphic solutions of some difference equations are studied,respectively,based on classical results and research of Nevanlinna theory.The structure of this dissertation is as follows:In Chapter 1,the background and domestic and overseas research status of this topic are introduced.At the same time,the main research work of this dissertation is given.In Chapter 2,some definitions and theorems of Nevanlinna theory,some results of difference value distribute are given to prepare for following chapters.In Chapter 3,some properties of exponential polynomials are used in this dissertation to investigate the growth of solutions of the following complex second order linear differential equation f"+A(z)f’+B(z)f=0,(0.0.1)where A(z)and B(z)((?)0)are entire functions.Some conditions which every nontrivial solution of the equation(0.0.1)is infinite order are obtained,respectively,based on A(z)is a transcendental entire function and B(z)is a nonconstant polynomial.In Chapter 4,first,the[p,q],φ order and[p,q],φ type which measure the growth of meromorphic functions are defined,then the growth relationships between coefficients and solutions of the complex higher order linear differential equation(0.0.2)is found by using the concepts of[p,q],φ order and[p,q],φ type.f(k)+Ak-1(z)f(k-1)+…+A1(z)f’+A0(z)f=0,(0.0.2)where Aj(z)(j=0,1,...,k-1)are entire functions,In Chapter 5,the properties of solutions of the following third-class difference equations are discussed,respectively,based on results of difference analogue.and where q be a finite non-zero complex number,cj(z)are rational functions,P(z,f(z))and Q(z,f(z))are relatively prime polynomial in f(z).Then,the conditions which the above equations has no transcendental meromorphic solution is obtained,and the lower bound of maximum module of solution the equation(0.0.3)and(0.0.5)which admit a transcendental entire solution is obtained,and the pole number of solution of equations which admit a meromorphic solution is obtained.In the last,the research content and methods of this dissertation are summarized,and further,some future research topics are expected.