Uniqueness Of Meromorphic Functions Related To Their Difference Operators

In1920s, R.Nevanlinna, a famous Finnish mathematician, established Nevanlinnatheory, which is called the first and the second Nevanlinna basic fundamental theorem.The theory is surely one of the most important achievement in mathematics in the20thcentury and is the important tool of theoretical research complex. In about half acentury, Nevanlinna theory has been well developed and can be used in the researchof the complex differential equation, the uniqueness of meromorphic function andother fields. The research of the uniqueness theory of meromorphic function involvingpublic values originated in R.Nevanlinna some work （see e.g.[32]）, R.Nevanlinnaused he first and the second Nevanlinna basic fundamental theorem he established toobtain two famous Nevanlinna’s five-value theorem and Nevanlinna’s four-valuetheorem, which is the basis for the uniqueness of meromorphic functions.From the end of the1950to the begin of1960s, Qiong Qinglai and Yang Le, theolder mathematician in our country, made some profound results in this respect. In1980s, famous mathematician F. Gross G. G. Gundersen M.Ozawa G.Frank E.Mues N. Steinmetz W. Bergweiler Hong-Xun Yi and so on obtained manyimportant research results in the uniqueness theory of meromorphic function. In1995,Hong-Xun Yi solved the Gross question which was not solved in the past twenty years.which plays a role in the development of the uniqueness theory of meromorphicfunction.Recently, Yik-Man Chiang Shao-Ji Feng R.G.Halburd R.J. Korhonen I.Laineand so on, have founded Nevanlinna characteristic of difference, Nevanlinna theoryfor the difference operator, the value distribution theory of difference polynomial anddifference analogue of the lemma on the logarithmic derivative, To study theproperties of the solutions of the difference equation and the difference uniqueness theory to lay the foundation （see e.g.[6][7][8][16][17]）.The present dissertation is the author’s research work under the cordial guidanceProfessor Xiao-Min Li. It consists of three chapters.In chapter one, we briefly introduce some main concepts, usual notations, classicalresults in the value distribution theory of meromorphic functions and the results ofdifference Navenlinna theories.In chapter two, we study the uniqueness problem of entire function sharing a smallentire function with their difference operators, the results in this paper improve thosegiven by Kai Liu and Lian-zhong Yang.Now we turn to state the main results of the thesis.Theorem1Let f be a non-constant entire function such that p（f）＜2,Let be a nonzero complex number,and let a be non-vanishing entire function suchthat p（a）＜p（f） and λ（f-a）＜p（f）. Then we have f-a与and△_ηⁿ f-a share0CM, and△_η²n a （z）-△_ηⁿ a （z）=0, for all z∈C, where A, Bare nonzero constants and e^Az=1.Theorem2Let f be a non-constant entire function, Let P be a Polynomialsuch that and λ （f-P）＜p（f）＜2, let η be a nonzero complex number. Then wehave: f-P and△_ηⁿ f-P share0CM, if and only Ifand△_ηⁿ p （z）-△_ηⁿ p （z）=0for all z∈C, where are nonzeroconstants and.e^Aη=1In chapter three, we study the uniqueness problem of non-constant meromorphic functions sharing three values with their difference operators, and obtain one maintheorem which is improvement of results given by J.Heittaokangas, R.Korhonen,I.Laine and J.Rieppo.Now we turn to state the main result of the thesis.Theorem3Let f be non-constant meromorphic function of a finite order,and let η be a nonzero complex number. If f and△_ηfshare a₁,a₂,a₃CM, wherea₁, a₂,a₃are three distinct values in the extended plane. Then2f （z）=f （z+η）forall z∈C.