Results On Value Distribution Of Meromorphic Functions Involving Difference Operators And Algebroid Functions

The Finland mathematician R.Nevanlinna made a great contribution to the establishment of the distribution value of meromorphic functions.In the 1920s,he noted characteristic function that can describe the growth of mero-morphic functions naturally,and two very important theorems,which are called the first and second fundamental theorem.The theory not only has a milestone significance for the value distribution of meromorphic functions,but also became an indispensable and strong tool for complex analysis.There is a close connection between uniqueness and sharing values.Early in 1926,Nevanlinna proved the four value theorem and five value theorem ac-cording to his value distribution theory of meromorphic functions.During the following decades,more and more mathematicians are starting to get involved in the area,and many uniqueness theorems of meromorphic functions or en-tire functions emerged.Nevanlinna theory continuously deepened,matured and enriched,and has been successfully applied in many other related fields,for instance,dynamic system,complex differential equations,analytic number theory,and multiple complexes.The earth has promoted the development of mathematics greatly.In the past century,there are many interesting,simple and perfect results in the uniqueness of meromorphic functions.It is very important to intro-duce difference operators,algebroid function and derivative,and combine the functions themselves to study value distribution.There are many good conclu-sions at present,but still have lots of problems unsolved or need to be further improved.I carefully read a lot of references about meromorphic functions involving difference operators and algebroid functions under the guidance of professor Peichu Hu.I carried out the research of their value distribution on the basis of the predecessors,mainly investigate applications in meromorphic functions involving difference operators and algebroid functions,improve and generalize some known results.At first,for the value distribution of meromorphic functions with their k-order derivatives,Hongxun Yi(see[17])proofed a theorem in 1994,and obtain a corollary of entire functions.In 2015,Cuiping Zeng(see[18])gener-alized this result to difference operators of finite order meromorphic functions.Considering a more general difference operators,she replaced k-order deriva-tives shared value by k-order difference operator shared value,and got a corollary of entire functions.In this paper,we weaken the condition of theorem in[18],proof thatTheorem 1 Let f(z)and g(z)are meromorphic functions,and A(f)<?.F(z)and G(z)are difference operators of f(z)and g(z)respectively,and F(z)? C,G(z)? C.If F(z)and G(z)share 1 CM,f(z)and g(z)share 0 CM,and N(r,f/1)+N(r,g/1)+(3k-1)N(r,f)<(?+o(1))T(r),where 0<?<1,T(r)=max{T(r,f),T(r,g)}.Then we have F·G = 1 or F?G.The theorem expands the scope of k,thus improves the result in[18].At this point,he corollary 1 in[18]still valid.For the uniqueness of algebroid functions,Ulirch,Valiron,Eremenko and Yuzan He has made many perfect conclusions.In 2014,Zongsheng Gao and Yunbo Jiang(see[37])studied the uniqueness of algebroid functions with their derivatives.In this paper,we replace the condition“share 0 CM”by“share an arbitrary finite complex number CM",prove the following conclusion.Theorem 2 Let w(z)be the v-valued algebroid function with v>2,w and w' share 2v distinct finite complex numbers b1,b2,…,b2v CM.Then we have w = w' if there exist two nonnegative real numbers c,R such that v single-valued analytic branches w1,w2,…,wv of w satisfy |wj(z)|>c and|wj/(z)|>c(j = 1,2,…,v)for |z| = r>R.Because the main difficulties in the study of algebroid functions is their branch points,so if we consider a class of special algebroid functions which branch points growth slowly,namely satisfy Nx(r,w)= o(T(r,w)),will sim-plify the research difficulty.And for the special functions,Huifang Liu proofed a uniqueness theorem(see[34])in 2011:Let w(z)be the v-valued algebroid function,and Nx(r,w)=o(T(r,w)),b1,b2 are two distinct nonzero finite complex numbers.If w and w'share 0,b1,b2 CM,then we have w = w'.Here,we can weaken the condition of this result as follows,prove the condition“CM shared b1,b2" can be replaced by“IM shared b1,b2 ".Theorem 3 Let w(z)be the v-valued algebroid function such that w and w' share 0 CM,and share two distinct nonzero finite complex numbers b1,b2 IM(ignoring multiplicities).Then we have w = w' if Nx(r,w)= o(T(r,w)).From the proof of this theorem,it can be seen that the condition "CM sharing 0" plays a very important role.We further proved that the special con-dition "CM sharing 0" can be replaced by "IM sharing three finite nonzero complex numbers”.Theorem 4 Let w(z)be the v-valued algebroid function such that w and w' share five distinct nonzero finite complex numbers b1,…,b5 IM.Then we have w = w' if Nx(r,w)= o(T(r,w)).Obviously,Theorem 3 and Theorem 4 improve and generalize the result of Huifang Liu.The dissertation is structured as follows:In chapter 1,we briefly introduce some basic knowledge,the basic con-cepts,notations,and some primary results of Nevanlinna theory;In chapter 2,we investigate value distribution of finite order meromorphic function involving difference operators,and generalize a conclusion of Cuiping Zeng;In chapter 3,we research the uniqueness of algebroid functions with their derivatives.Firstly,we improve a result of Zongsheng Gao and Yunbo Jiang,then consider a class of special algebroid functions,and generalize a theorem of Huifang Liu.