Normality Criterion Of Families Of Meromorphic Functions With Multiple Valueand Derivative

Since 1907 the concept of normal family has been proposed by P.Montel ,the theory ofnormal family has become an important direction in complex analysis,then finding newcriterions of normal family is being a significant problem.Many scholars gain fruitfulachievements on it.Many facts are related to the criterions of meromorphic functions, in thisthesis,the both facts that multiple value and derivative are discussed in establishing newcriterion of normal family.New criterions of normal family are worked out in this thesis,which is composed of fourchapters.In the first chapter general background and research proceedings of theory of normalfamily are introduced, the necessary basic knowledge and some important conclusions ofnormal family are showed also,then proof of the result1.5.6 are arranged in the last section ofthis chapter.In the second chapter,a new criterion of normal family is proved, which under thefollowing conditions: for every function in the meromorphic family,of which the multiple ofzeros at least k + 2 and the poles at least 2,the derivative function with k-order doesn't take ameromorphic function which isn't always equivalence zero.This criterion is established on thebase of the proved one,it's about the limiting condition on the multiple of zeros and poles ,andthe derivative function with k-order.In the third chapter,a new criterions of normal family with more complicated high-orderdifferential rational function are found but no other condition is changed,it also come with thesame conclusion with.In addition,a theory about value distribution with high-order differentialrational function is confirmed too,which is very useful in the proof process.In the last chapter,one criterions of normal family is given on the basis of the thirdchapte,which is extension of Hayman conjecture to some extent,and with the view of sharevalue problem,another new criterion of normal family is obtained too.