Criteria For Normality Of Meromorphic Functions With Shared Values

As early as 1907, P.Montel introduced the concept of a normal family of functions as he called so with considering a certain compactness of the family. Over the past few decades, scholars have obtained many profound results on the normality of meromorphic functions. In this thesis, based on the proceeded research, we use Nevanlinna’s theory of value distribution and Zalcman-Pang technology to study the criteria for normality of meromorphic functions with shared values, and obtain some corresponding results.First, we discuss the criteria for normality of families of meromorphic functions involved with moving targets shared by differential monomials, and we obtain the following results.Theorem 2.1.1 Let a(z) be an entire function without zeros, k≥3 a positive integer, and F a family of meromorphic functions on a domain D, where the zero’s multiplicity of each function f in F is at least k and the pole’s multiplicity of f is at least 2. If for each pair of f,g ∈ F, ff(k) and gg(k) share a(z), then F is normal on D.Theorem 2.1.2 Let a(z) be an entire function without zeros,k≥2 be a positive integer, and F be a family of meromorphic functions on a domain D, where the zero’s multiplicity of each function f in F is at least k and the pole’s multiplicity of f is at least 3. If for each pair of f,g ∈ F, ff(k) and gg(k) share a(z), then F is normal on D.Next, we investigate some problems on seeking unnormal points for a family of meromorphic functions of which every element share a pair of functions with its kth derivatives, and obtain some corresponding results.Theorem 3.1.1 Let k≥2 be a positive integer, a and b be two holomorphic functions not vanishing identically on a domain D, and F be a family of meromorphic functions on D, where the zero’s multiplicity of each function f in F is at least k+1 and for each f∈F, f=a(?)f(k)=b. If F is informal at z0∈D, then z0 must be the zero of φ(z).Theorem 3.1.2 Let a and b be two holomorphic functions not vanishing identically on a dmain D such that a doesn’t take the value 0 at the zeros of b . Let F be a family of meromorphic functions on D, where the zero’s multiplicity of each function f in F is at least 2 and for each f∈F, f=a(?)f=b.If F is informal atz0∈D, then z0 must be the zero of φ(z).Theorem 3.1.3 Let k be a positive integer, (p(z) be a holomorphic functions not vanishing identically on a domain D, and F be a family of meromorphic functions on a domain D, where the zero’s multiplicity of each function f in F is at least k+1 and the pole’s multiplicity of f is at least2, for each f∈F,f(k) (z)≠φ(z). If F is informal at z0∈D, then z0 must be the zero of φ(z).