Some Researches On Shift Operators?Difference Operators And Differential Operators Of Meromorphic Functions

Since the appearance of the value distribution theory of meromorphic function-s in 1925 which was established by R.Nevanlinna,a mathematician in Finland,the uniqueness of meromorphic functions remains as an important and interesting branch of complex analysis until now.This dissertation is contributed to study the uniqueness on periodic meromorphic functions,involving the related uniqueness of shift operators?difference operators and differential operators.The structure and results of this dissertation are organized as follows.In chapter 1,we briefly introduce the value distribution theory of meromor-phic functions,the uniqueness theory of meromorphic functions,and the difference analogues of value distribution theory of meromorphic functions.In chapter 2,we firstly prove an uniqueness theorem for meromorphic functions of hyper-order less than 1 and their shift operators concerning "2CM + 1IM" shared values,and show the precision and necessity of their conditions through examples.This theorem extends a result of meromorphic function theorem due to Heittokangas el ta from finite order to infinite order.Meanwhile,it examplifies the precision and necessity of the conditions therein.Secondly,we obtain some results on certain meromorphic functions which partially share or truncated-share 2 or 3 values with their shift operators to partly answer the open question of“1CM+2IM".Also,examples show that the conditions of those results are necessary.In chapter 3,we firstly prove a periodicity theorem of meromorphic functions,which completely improves a result due to Brosch from "3CM" to "2CM +1IM"and is precisely illustrated by examples.Secondly,by taking advantage of the new characteristics of value distribution of perio6dic meromorphic functions,we obtain an uniqueness theorem related to periodic meromorphic functions,which complete-ly improves a result due to Zhen Jianhua from "3CM" to "2CM + 1IM".Also,examples are given to show the precision of this result and the necessity of the con-ditions.Finally,we obtain some new theorems of certain meromorphic functions and periodic meromorphic function,both of which have at most finitely many poles,concerning three shared-values or truncated-shared-values.Beside,examples show that the conditions of those results are also necessary.In chapter 4,basing on the proper auxiliary functions and the second main theorem of the difference analogues of value distribution theory of meromorphic functions,we prove an uniqueness theorem of meromorphic functions of hyper-order less than 1 and their difference operators with some partially shared values,which af-firmatively answer a conjecture raised by Chen-Yi.Moreover,the shared conditions of our result are more general than the shared conditions of Chen-Yi's conjecture.In chapter 5,we obtain the expressions of meromorphic functions of zero order which share one value or one set with their differential operators on the basis of the uniqueness of the coeffcients of the Laurent expression.Our results are the supple-mentaries of some related results due to Li-Yi and Qi el ta.Moreover,examples are also given to illustrate that the conditions are necessary.