| In this thesis, by using Navanlinna theory, theory of normal families and Wimann-Varilon theory, we will prove some results about the growth of meromorphic solutions of algebraic differential equations, the primeness of some functions. We will also prove some theorems in uniqueness theory. Finally, we study the zeros of derivatives and homogeneous differential polynomials of meromorphic functions;In Chapter 2, we will first give a normal criterion relating to algebraic differential equations and discuss the growth of the specially entire solutions of some algebraic differential equations. Secondly we consider the following differential equations: Cz,ww '2+B z,ww'+Az, w=0. We will prove that (1) under some conditions, the order of any entire solution of these equations must be no less than 1. (2) If C(z, w) ≠ 0, then the order of any transcendental meromorphic solutions of these equations must be ½n, where n is a nonnegative integer, conversely, any such number ½n is the order of a transcendental meromorphic solution of a certain equation of the above form. Then the growth of the meromorphic solutions of the equation f''=Lz,f f'2 +Mz,ff'+N z,f, where L, M, N are birational functions, is studied. We will prove that if L(z, f) satisfies a quite general condition, then f must be of finite order. Furthermore, if L(z, f) ≡ 0, and M(z, f), N(z, f) are polynomials in f, then the order of any entire solution of the equation is a positive integral multiple of ½, there is an example to show that this is not true for meromorphic solutions. We will discuss the growth of meromorphic solutions of Schwartz differential equations in the last section of this chapter.;In Chapter 3, we will prove that Gamma function is prime. We will give an estimation of the Nevanlinna characteristic function of Riemann zeta function, then by using this, we will prove that Riemann zeta function is prime. We will prove that all transcendental entire solutions of first-order algebraic differential equations with rational function as coefficients are pseudo-prime.;In Chapter 4, we will give a certain kind of unique range set for meromorphic functions and discuss the functions which share some values with Gamma function.;In Chapter 5, we will show that if F is a meromorphic function of finite order such that either F has only finitely many simple zeros and F″ has only finitely many zeros or F(z)F ″(z) - h(z) F'(z)2 has only finitely many zeros, where h(z) is a polynomial and hz≢1 and hz≢n+1 n , then Fz=Rz ePz, where R(z) is a rational function and P(z) is a polynomial. This generalizes a result due to Bergweiler. |
