Uniqueness Of Meromorphic Functions Sharing Values And Its Applications

The uniqueness theory of meromorphic functions mainly studies conditions under which there is only one function. We know that the conditions used to determine a transcendental meromorphic function is different from that of a polynomial. Therefore, uniqueness theory on meromorphic functions becomes important, complicated and interesting. The theory dealing with shared values originates from some works in R. Nevanlinna. These works lay the foundation of the research on the uniqueness theory, which has quite an effect on the development of mathematical branches, and on the interaction among them. In recent decades, many mathematicians paid close attention to it, which makes it become a very active subject all around the world. Many new results appear as time goes on. On the basis of Nevanlinna's 5IM theorem and 4CM theorem, a lot of deep results were obtained by Xiong Qinglai and Yang Le. Many other mathematicians such as F. Gross, W. Hayman, et al, also obtained a lot of elegant results on the research of the uniqueness theory.In 1994, Prof. Yi Hongxun completely solved a famous problem proposed by F. Gross which promotes the research on the uniqueness theory. In this paper, we will give some results on the uniqueness theory of meromophic functions which consist four chapters under the guidance of my teacher LüWeiran.In chapter one, we briefly introduce some main concepts, fundamental results and usual notations concerned with this thesis in the value distribution theory of meromorphic functions.In chapter two, we investigated the growth of solutions of an inhomogeneous differential equation, and prove the following result.Theorem A The order of any non-zero solution f ( z )of the differential equation ( z)satisfyingρ( f)=∞, where p1 ( z )is a transcendental entire function of order less than1/ 2, p2 ( z )is a transcendental entire function of order less than1.In chapter three, we study the uniqueness problems on meromorphic functions concerning differential polynomials. We improve the results given by S.S. Bhoosnurmath, R. Dyavanal and X. Zhang, W. Lin, and obtain the following results.Theorem B Let f ( z )and g ( z )be two non-constant transcendental meromorphic functions, and let n , k ,m be three positive integers that satisfying n > 9 k + 6 m*+ 13. And suppose that [ f n ( z )(μf m ( z) +λ)]( k)and [ g n ( z )(μg m ( z) +λ)]( k)share 1 IM, hereλandμbe two constants such that|λ| + |μ|≠0, and f ,g share∞IM, then (1) Ifλμ≠0, when m > 1and ( n , n + m) = 1, then f≡g, and when m = 1and ( , f)2Θ∞> n, then f≡g. (2) Ifλμ= 0, then either f = tg, where t is a constant satisfyingTheorem C Let f ( z )and g ( z )be two transcendental meromorphic functions, and let n , k ,m be three positive integers satisfying n > 9 k + 4 m+ 15. Suppose that [ f n ( z )( f ( z ) ? 1) m ]( k)and [ g n ( z )( g ( z ) ? 1) m ]( k)share 1 IM, and f ,g share∞IM, then either f≡gor f ,g satisfying R ( f , g )≡0, where R (ω1 ,ω2 ) =ω1 n (ω1 ? 1) m ?ω2n (ω2? 1) m .In chapter four, we consider the uniqueness problem of meromorphic functions concerning their derivatives sharing one value and obtain some results which improve some known theorem.Theorem D Let F ( z )and G ( z )be two non-constant meromophic functions, andk be a positive integer. If F ( k)and G ( k)share1CM, and Then F ( k ) G ( k)≡1or F≡G.Theorem E Let F ( z )and G ( z )be two non-constant meromophic functions, andk be a positive integer. If F ( k)and G ( k)share1IM, and Then F ( k ) G ( k)≡1or F≡G.