Uniqueness And Normal Family Of Meromorphic Functions Concerning Shared Values

The uniqueness theory of meromorphic functions is an important subject in meromorphic function theory. It mainly studies conditions under which there exists essentially only one func-tion satisfying these conditions. Nevanlinna established the value distribution theory in1920s, Five-value Theorem and Four-value Theorem are obtained by Nevanlinna via his founding value distribution theory. Later, many scholars weakened the conditions and obtained a lot of impor-tant results.In the complex plane a transcendental meromorphic function assumes any complex num-ber infinitely many times with at most two exceptional values, but in an angular domain this conclusion is not always right. As the existence of Borel directions, many scholars began to study the uniqueness of meromorphic function in an angular domain. Zheng JianHua extended the Five-value Theorem to an angular domain. Zhang QingCai extended Four-value Theorem to an angular domain via the Ahlfors-Shimizu Characteristic. But there are still many interesting open problems to study.Montel introduced the conception of normal family, he defined a set of functions with some kind of compactness as a normal family in infancy century. The great development of normal family theory is due to its combination with Nevanlinna theory. Zalcman gave a necessary and sufficient condition that makes meromorphic abnormal in1975. Chinese mathematicians connected Zalcman’s results with the function derivative, a series of new normal criterions were set up. In1990s, Schiwick put forward an idea that combined normal family and uniqueness together, in this field Chinese and Israel mathematicians have done the main work.In this paper, we mainly research the uniqueness concerning shared values and normal family of meromorphic functions sharing functions. The full text is divided into four chapters.In Chapter1, we outline some basic concepts, results and notations of Nevanlinna value distribution theory; the Nevanlinna Characteristic, the Ahlfors-Shimizu Characteristic; normal family and normal criterions etc.In Chapter2, we first research if for any value a≠0, transcendental meromorphic functions f(z) and g(z) satisfy Ep)(a,[fnP(f)](k)=Ep)(a,[gnP(g)](k), then fnP(f)=gnP(g), which greatly improves the conclusion of zhang, chen and Fang. And we research the case that transcendental meromorphic functions f(z) and g(z) share∞IM, then f=tg, where tn+m=1; or f(z)=C1ecz, g(z)=C2e-cz, where c1, c2, c and am satisfy (-1)ka2m(c1C2)n+m c2k(n+m)2k=a2.In Chapter3, We investigate the uniqueness problem of the finite or infinite order of mero-morphic functions f(z) and g(z) share three values in an angular domain, and obtain f=g or fg≡1.In Chapter4, We mainly investigate some normal criterions of meromorphic family F, if for (?)(f, g)∈F, ff(k) and gg(k) share a holomorphic function p(z) in a domain D, then F is normal in D. And if for (?)(f, g)∈F, P(f)f(k) and P(g)g(k) share a holomorphic function p(z) in D, then F is normal in D.