Some Applications Of The Value Distribution Theory

In this paper, we mainly investigate some applications of Nevanlinna theory of the value distribution of meromorphic function, including some problems of zeros and the value distribution of a type of meromorphic function, and the existence of admissible meromorphic solutions of types of systems of higher-order algebraic differential equations in the complex domain.The paper contains four chapters. We mainly introduce the basic definitions and fundamental theorems and a few commonly used notations of Nevanlinna theory of the value distribution in the first chapter.In chapter two, we investigate the problem of the value distribution of a type of function inthe form of f^m(f^k)ⁿ, and discuss the more precision quantitative estimation for the number of zeros of f^m(f^k)ⁿ -(?)(z). We come to the conclusion of boundary of the characteristic function T(r,f) of the transcendental meromorphic function f(z).With different methods from that of chapter two, in chapter three we further investigate the problem of zeros of f^m(f^k)ⁿ -(?)(z) testifies that if k≥1,n,m≥2 ,the function off^m(f^k)ⁿ-(?)(z)has infinitely zeros, and thus it improves and generalizes some relative resultson this topic.In chapter four, in virtue of Steinmetz' s method and Nevanlinna theory of the value distribution of meromorphic function, we investigate the existence of admissible meromorphic solutions of a type of systems of algebraic differential equations in the complex domain. We obtain an analogous result similar to the Malmquist type Theorem of the generalized algebraic differential equation.