| In 1920s, R. Nevanlinna introduced the characteristic functions of mero-morphic functions and gave the famous Nevanlinna theory which is one of the greatest achievements in mathematics in the 20th century. The Nevanlinna the-ory also be called the value distribution theory of meromorphic function. This theory is composed of two main theorems, which are called Nevanlinna’s first and second main theorems that had been significant breakthroughs in the develop-ment of the classical function theory, since the latter generalizes and extends the Picard’s theorem greatly, and hence it denoted the beginning of the theory of meromorphic functions. Moreover, Nevanlinna theory is considered to be basis of modern meromorphic function theory, and it has a very important effect on the development and syncretic of many mathematical branches, for example, po-tential theory, complex difference equations, normal family, over certain complex manifolds and so on.In 1907, P. Montel introduced the concept of normal family. It plays an im-portant role in complex dynamics. A family （?） of meromorphic functions defined in domain D is said to be normal, in the sense of Montel, if for any sequence {fn}（?）<（?）, there exists a subsequence{fnj} such that fnj converges spherically locally and uniformly in D, to a meromorphic function or oo. （see[44],[53]）. Our aim is to find the normal families. One guiding principle in the study has been the heuristic principle which says that a family of functions meromorphic （or holom-phic） in a domain and possessing a certain property is likely to be normal if there is no nonconstant function meromorphic （or holomphic） in the plane which has this property. This is so-called Bloch’s principle. Indeed, Bloch’s principle is not true in general, but it is still an important guiding principle in the theory of normal family. Many Chinese mathematicians, such as Prof. L. Yang, Prof. G. H. Zhang, Prof. Y. X. Gu, Prof. H. H. Chen, Prof. X. C. Pang, Prof. M. L. Fang, Prof. J. M. Chang, etc, have made remarkable works and contributions to the development of the normal family theory in the world. The research on the normal family theory of meromorphic functions is a very active international sub-ject in recent decades, especially combing Zalcman-Pang’s Lemma with sharing values. After that, a lot of elegant results were given by many mathematicians.In 1929, R. Nevanlinna applied the value distribution theory to consider the conditions under which a meromorphic function of a single variable could be determined and derived the famous Nevanlinna’s five-value and four-value theo-rems. From then on, many foreign and domestic mathematicians have devoted themselves to the research and obtain lots of elegant results on the research of the uniqueness theory, such as F. Gross, M. Ozawa, G. Frank, E. Mues, N. Steinmetz, H. Ueda, G. Gundersen, Q. L. Xiong, L. Yang, etc. In the past two decades, Pro-fessor H. X. Yi did much creative work on the uniqueness theory of meromorphic functions, and well improved the development of the uniqueness theory.The present thesis involves some new results of the author, under the guid-ance of my supervisor. The dissertation is composed as follows.In chapter 1, we introduce the general background of Nevanlinna Theory, the development of the Normal Family Theory.In chapter 2, we investigate the uniqueness problem of entire functions that share polynomials or rational functions with their derivatives. By estimating the size of pn’s in the famous Zalcman-Pang’s lemma and using the theory of normal family, we deduce this kind of functions have finite order, which is an important property. And then, we obtain some uniqueness theorems, which improve the results given by L. A. Rubcl and C. C. Yang （see[42]）, J. T. Li and H. X. Yi （see[24]）. Meanwhile, some examples show that the conditions of the results are necessary. In fact, we （see[38]） obtained the following results.Theorem 0.1. Let Q1（z）=a1zp+a1p1zp-1+…+a1,0 and Q2（x）=a2zp+ a2,p-1zp-1+…+a2,0 be two polynomials such that deg Q1（z）=deg Q2（z）=p （where p is a non-negative integer） and a1, a2（a27（?） 0） are two distinct complex humbers.Let f（z）be a transcendental entire function.If f（z）=Q1（z）=f1（z）= Q1（z）and f1（z）=Q2（?）=Q2（z）,then f（z）（?）f1（z）.Theorem 0.2.Let Q1（z）=a1zp=a1,p-1zp-1=…=a1,0 and Q2（z）=a2zp+ a2,-1zp-1+…+a2,0 be two polynomials such that deg Q1（z）=deg Q2（2）=p （where p is a non-negative integer）and a1,a2（a2（?）0）are two distinct complex numbers.Let f（z）be a non-constant entire function,and f（z）=Q1（z）（?）.f1（z）= Q1（z）and.f1（z）=Qz（z）（?）f（z）=Q2（z）,then f（z）if of finite orderWe also study the uniqueness problem of entire functions share rational func-tions with their derivatives and obtain some results（see[27]）which improve some known theorems of J.T.Li,H.X.Yi（see[24]）and J.M.Qi,F.Lu,A.Chen （see[38]）.We obtain:Theorem 0.3. Let f be a transcendendental meromorphic function uith funitely many poles,and let R1 and R2 be two distinct rational functions such that where M is a positive constant.If（1）f and R1 have no common pole,（2）f（z）=R1（z）（?）f’（z）=R1（z）and（3）.f1（z）=R2（z）（?）.f（z）=R2（z）,then f（z）（?）f’（z）.Theorem 0.4. Let f be a transcendental meromorphic function with finitely many poles,and let R1 and R2 be two distinct rational functions such that where M is a positive constant.If（1）f（z）=R1（z）（?）f’（z）=R1（z）and（2）f’（z）=R2（z）（?）f（z）=R2（z）, then f is of finite order.In chapter 3,we study some normal criteria of meromorphic function about shared values with their multiplicity zeros.Our results（see[31]）improve some obtained results of Y. F. Wang, M. L. Fang （see[48]）, M. L. Fang, L. Zalcman （see[10]）. In fact, we （see[39]） obtained the following results.Theorem 0.5. Let k（k≥2） be an integer and b be a nonzero finite complex number, and let （?） be a family of meromorphic functions in domain D, all of whose zeros have multiplicity at least k+2. If, for every pair f, g∈（?）, all zeros of f（k）（z）,g（k）（z） are multiple, f（k）（z） andg（k）（z） share b in D, then （?） is normal in D.Theorem 0.6. Let （?） be a family of meromorphic functions in domain D, all of whose zeros are multiple and let n（n≥2） be an integer and a, b be two nonzero finite complex numbers. If f+a（f1）n and g+a（g1）n share b in D for every pair of functions f,g∈（?）, then （?） is normal in D.In chapter 4, we do some research on the results of Goldberg’s theorem concerning the growth of meromorphic solutions of algebraic differential equations or a type systems of algebraic differential equations. This work improves some very recent results of W. J. Yuan, B. Xiao, J. J. Zhang, （see[55]） and R. M. Gu, J. J. Ding, W. J. Yuan, （see [14]）.Theorem 0.7. Let w（z） be a meromorphic function in the complex plane and all zeros of w have multiplicity at least k （k∈N）, P[w] be a polynomial with the form （4.1.2）（see chapter 4） and nkq>deg P[w] （n∈N）. If w（z） satisfies the differential equation [Q（w（k）（z））]n= P[w], then the growth orderλ:=λ（w） of w（z） satisfies where Q（z） is a polynomial with degree q.Theorem 0.8. Let w=（wi,w2） be the meromorphic solution of a type systems of algebraic differential equations of the form （4.1.3）（see chapter 4）, if m1 m2qk>u, and all zeros of w2 have multiplicity at least k （k∈N）, then the growth ordersλ（wi） of wi（z） for i=1,2 satisfy where a=deg（a（z））m2, b=maxj∈Imax{deg bj（z, w2）,0}. In the finial chapter 5.we investigate a special differential equation and obtain some results which improve the conclusions proved by P.Li（see[31]）.Theorem 0.9.Let n≥2 be an integer.Let f be a transcendental entire function, P（f）be a differential polynomial in f of degree≤n-1.If where Pi（i=1,2）are nonvanshing small functions of ez,Q1（z）=αkzk+αk-1zk-1+…+α1z+α0,Q2（z）=βkzk+βk-1zk-1+…+β1z+β0 are two polynomials satisfying（n-1）βk≥nαk>0（αk-1…α0,…β0 are finite con-stants）and k≥1 is a positive integer,then there exists a small functionγof f, such thatTheorem 0.10.Let n≥2 be an integer and P1,P2 be small functions of ez.If there exists a transcendental entire function f satisfying the differential equati.on （0.1）.where P（f）is a diffrential polynomial in of degree not exceeding n-2 andαk<0<βk,thenαk+βk=0,and there exist constants c1,c2 and small functionsβ1,β2 with respect to f such that moreover,βin=Pi,i=1,2.Theorem 0.11.Let n≥2 be an integer,P1,P2 be small functions of ez. If（?） is irrational,then the differential equation（0.1）has no entire solutions, where P（f）is a differential polynomial in f of degree≤n-1 and（n-1）βk≥nαk>0. |
