| he uniqueness theory of meromorphic functions studies the uniqueness conditions which can, to some extent, uniquely determine a meromorphic function. There are close relationship between uniqueness and sharing values. Early in 1926, R. Nevanlinna proved the four values theorem and five values theorems according to his value distribution theory of meromorphic functions. Since then, for about forty years, there appeared no any interesting results on value sharing of meromorphic functions or entire functions till 1970s, Gross, Yang, Ozawa, etc. started the research of this field. They and their follows have been published many papers in uniqueness theory and value sharing by combining the classical function theory and the value distribution theory of meromorphic functions. Moreover, the concept of sharing values has been extended to that of sharing sets by Gross and Yang in 1970s. They also introduced the concept of unique range set of entire functions or meromorphic functions, identified by URSE (URSM). The smallest cardinalities of a URSE or URSM becomes an interesting research problem and several related results were obtained by H. X. Yi, E. Mues, and Li-Yang.;The present thesis will present several generalizations of four values theorem and five values theorem by combining sharing values, sharing sets and sharing pairs. For instance, we prove that if two nonconstant meromorphic functions f and g share four values and another finite set IM, then f and g share all values CM, thus g is a Mobius transformation of f. On the unique range set of meromorphic or entire functions, we will prove that the smallest cardinality of unique range sets is great than or equal to five. As a related topics, we have defined a new concept of uniqueness polynomial of meromorphic or entire functions and obtain a necessary and sufficient condition for a polynomial of degree 4 to be a uniqueness polynomial. Some questions deal with an entire function and its linear differential polynomial share two values IM also have been studied. Particularly, we present an affirmative answer to Frank's conjecture. We also prove that if an entire function f share two finite values CM jointly with its derivative... |
