Some New Results About Value Distribution And Normal Family Theory For Meromorphic Functions

Normal family and value distribution theory for meromorphic functions are the main sub-jects of our study. The two are promoting the further development of each other. In the process of exploring, we obtain some new results which make breakthrough progress in the research field of the corresponding.1.Normal family for meromorphic functions and the sequence of omitted func-tions.Recently, research on normal family theory for meromorphic functions have a great break-through. In chapter2. we firstly study the normal criterion involving the sequence of omitted functions by means of mathematical induction, and we give an example to show that assumption of the omitted functions being different is essentially different from hypothesis that the omitted functions are the same. Our results are as followsLet{/n}be a sequence of meromorphic functions on a domain D, whose zeros and poles have multiplicity at least3. Let{hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that{hn} converges locally uniformly in the spherical metric to a function h which is meromorphic and zero-free on D. D. If f’n hn, then{fn} is normal on D.2. Normal family for meromorphic functions and the differential inequalities.In chapter3, we mainly focus on the relationship between the differential inequalities and normality. A natural point of departure for this subject is the well-known theorem due to Marty. Recently, there has been activity in the study of differential inequalities with the reversed sign ("≥") of the inequality. Fortunately, we have the following interesting resultsLet k≥0be an integer, C>0, a>1are constants. Let F be a family of meromorphic functions in D. If for each f∈F,z∈Dthen F is normal in D.3. Normal family for holomorphic functions and the omitted function.Studying on the normal criterion, we ultimately want to consider the function family is a family of meromorphic functions. But some normal criterions apply only to the family of holomorphic functions. In chapter4, we continue to study the criterions which only apply to the family of holomorphic functions. We obtain a result where the omitted holomorphic function is improved to be meromorphic, and we give a counterexample to show that the result does not hold for a family of meromorphic functions.Let T be a family of functions holomorphic on a domain D C C, all of whose zeros have multiplicity at least k, wherek≥2is an integer. Let h(z)(?)0,∞be a meromorphic function on D. Assume that the following two conditions hold for every f∈F(a) f(z)＝0(?)|f(k)(z)|<|h(z)|and(b) f(k)(z)≠h (z).Then T is normal on D.4. One result value distribution theory for meromorphic functions.The application value of the normal family theory for meromorphic functions has a perfect show in the value distribution theory. In chapter5, we continue to study the Picard type theorem by using of the normal family theory and get a new Picard type theorem concerning elliptic function and high order derivative function.Let k>2be an integer, let h be a nonconstant elliptic function, and let f be a nonconstant meromorphic function in C, all of whose zeros have multiplicity at least k+1, except possibly finitely many. If T(r, h)=o{T(r,f)} as r�'∞, then f(k)=h has infinitely many solutions (including the possibility of infinitely many common poles of f and h).