Research On The Problems Of Normal Families And Value Distribution Of Differences Polynomials

In the prime of20century, R. Nevanlinna researched on meromorphic func-tions and introduced the characteristic functions of meromorphic functions. Then he gave the famous Nevanlinna theory which is one of the greatest achievements in mathematics in the20th century. The Nevanlinna theory is composed of two main theorems, which are called Nevanlinna's first and second main theorems, since the latter one extends the Picard's theorem greatly. The theory not only de-noted the beginning of the theory of meromorphic functions, but also is considered to be basis of modern meromorphic function theory. Now, Nevanlinna theory is applied in the fields of uniqueness theory, normal family, complex differential equation and complex dynamics etc..The recent papers [11] and [18] include independently obtained estimates for the proximity function m(r,f(z+c)/f(z)), when f(z) is a meromorphic function of finite order. The results may be viewed as discrete analogues of the lemma on the logarithmic derivative. On this basis, many scholars studied value distribution on difference operators.The theory of normal family is a important branch of complex analysis, in1907, P. Montel introduced the concept of normal family. It not only combined with value distribution theory, but also applied in complex dynamics which is a focus of the study now. Our aim is to find the criteria of normal families. Bloch's principle is not true in general, but it still plays an important roll in the theory of normal family. Many Chinese scholars contributed the advancement and devel-opment of normal family theory, for example: L. Yang, G. H. Zhang, Y. X. Gu, H. H. Chen, X. C. Peng, M. L. Fang and J. M. Chang and so on. The research on the normal family theory is a very active international subject in recent decades, especially applying Pang-Zalcman's Lemma and the ideas of sharing values. After that, a lot of elegant fruits were proofed by many mathematicians.The present thesis involves some new results of value distribution on differ-ences polynomial for meromorphic function and the normal family criteria with sharing values. The dissertation is composed as follows.In chapter1, we introduce the general background of Nevanlinna theory, the Nevanlinna theory on differences operators and the Normal family theory of meromorphic function.In chapter2, we investigate value distribution on differences polynomial of meromorphic function by the classic methods of uniqueness problem. Our meth-ods are different from ones in the previous theorems. In fact, we obtained the results on meromorphic firstly.Theorem0.1. Let f be a non-constant meromorphic function of finite order, s(z) be a small function of f(z). Suppose that P(z) is a polynomial, m is the cardinality of the set{z:P(z)=0} and deg(P(z))-m>3, then P(f(z))+f(z+c)-s(z) has at least one zero. If f is a transcendental merom,orphic function, then P(f(z))+f(z+c)-s(z) has infinitely many zeros.Theorem0.2. Let f be a transcendental meromorphic function of finite order, not of period c(≠0), and s(z) be a small function of f(z). Suppose that P(z) and m are as in Theorem0.1. If deg(P(z))-m>4, then P(f(z))+f(z+c)-f(z)-s(z) has infinitely many zeros.We also study the same problem on q-differences and obtain:Theorem0.3. Let f be a meromorphic function of zero order. q∈C\{0},.s(z) and P(z) satisfy the condition as Theorem0.1. Then P(f(z))+f(qz)-s(z) has at least one zero. If f is a transcendental meromorphic function, then P(f(z))+f(qz)-s(z) has infinitely many zeros.In chapter3, we study some normal criteria of meromorphic function about sharing differential polynomial. Our results improve some obtained results of C. L. Lei, M. L. Fang (see [32]).Theorem0.4. Let k be a, positive integer and let (?) be a family of meromorphic functions in the domain D with the multiplicity of all the zeros are at least k. Let P=apzp+…+a2z2+z be a polynomial. ap,a2≠0and p=deg(P)≥k+2. IF, for each f,g∈(?), P(f)G(f) and P(g)G(g) share a non-zero constant b IM in D. where G(f)=f(k)+H(f) be a differential polynomial of f satisfying w/deg|H≤k/l+1+1or w(H)-deg(H)<k, then (?) is normal in D.Theorem0.5. Let k be a positive integer, suppose that (?) be a family of mero-morphic functions in the domain D the multiplicity of all of whose zeros and poles are at least k and2respectively. Let P be a polynomial with two distinct zeros at least. If, for each f,g∈(?), P(f)G(f) and P(g)G(g) share a constant b IM in D. where G(f)=f(k)+H(f) be a differential polynomial of f with w(H)-deg(H)<k. then (?) is normal in D.In chapter4, we study some normal criteria of meromorphic function about sharing horomorphic function. Our results improve some obtained results of M. L. Fang, J. M. Chang (see [13]) and J. Y. Xia, Y. Xu (see [57]).Theorem0.6. Let (?) be a family of meromorphic functions defined in D, let ψ((?)0),ao, a1,,...,ak-1be holomorphic functions in D, and k be a positive integer. Suppose that, for every function f∈(?), f≠0, P(f)=f(k)+ak-1f(k-1)+...+a1f'+a0f≠0and for every pair functions (f,g)∈(?), P(f) and P(g) share ψ then (?) is normal in D.