Uniqueness Of Meromorphic Functions And Some Related Normality Problems

In1920s, R.Nevanlinna introduced the concept of meromorphic functionscharacteristic function and gave the famous Nevanlinna theory, that is, theory of valuedistribution of meromorphic functions on the complex plane. The theory is composed oftwo main theorems, which are called Nevanlinna’s first and second main theorems. Thetwo main theorems had been significant breakthroughs in the development of theclassical function theory, denoted the beginning of the theory of meromorphic functions,and made significant contribution on the development of other branches of mathematics.The Nevanlinna theory, by the self-improvement and development, is widely used inother areas of complex analysis, such as complex dynamic systems, complexdifferential equation, normal family as well as uniqueness of meromorphic function.This thesis, on the basis of the recent research，offers study of several aspects ofuniqueness of meromorphic function and normal family, which is divided into two parts.The first part is about the uniqueness sharing values of meromorphic functionsproblem. The uniqueness of meromorphic function is to investigate the only onefunction satisfies the given conditions under uncertain circumstances. As everyoneknows, polynomial is completely determined by zero point sets but a constant factor. Asfor the transcendental entire function and meromorphic functions, the factors are stillunknown. So how to uniquely identify a meromorphic function is complicated andinteresting. The uniqueness of family of meromorphic function is to determine theunique condition that can help with some common properties of a meromorphicfunctions. But for the share finite complexs value of meromorphic functions, which isalways a hot study point for uniqueness theory. Someone like Fang Mingliang supposethat differential polynomial of meromorphic function which has the same zeros and thesame special and multiplicity of zero must be the same as a series of harsh conditionsobtained two to meromorphic function must satisfy the relation. Based on these results,the author considers to generalize differential polynomial with special limitation ofmeromorphic functions into differential polynomials of meromorphic functions, andweaken the conditions that the multiplicity with the same zero points or the samenumber must be equal, i.e., to weaken the share condition and apply the theory ofWeight Share and truncate, which successfully gets relevant results. The results of thisthesis is an extension of Fang et al’s results. On the other hand, we make full use of the existing meromorphic function value distribution theory and differential operator indifference based on the Nevanlinna theory to study the uniqueness of holomorphicfunction difference.In the second part, we study normal families of meromorphic functions.Looking for new normal criterion is an important topic in the theory of normal family.In the researches of normal families a natural idea is to seek normal theorem from theuniqueness theorem (theorem of public value). Schwick took the lead in shared valuesand normal families of the combination and successfully got the normal families ofmeromorphic functions. Since then, many mathematical workers got many excellentresults on the basis of previous research. These results mostly relate to complex function.Our main work is to put to function of complex numerical problems existing in thenormal criteria conditions, for the promotion of functions holomorphic functions. Themain difficulty is that when the complex values replace all entire functions, it is hard todeal with the problem of zero. We use a variety of methods for normal family theory toovercome the difficulties in zero, successfully obtained several of meromorphicfunction normal criterion.