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

In this paper, we mainly study the value distribution and normal family theory for meromorphic functions, and get some new results for them. These results deeply improved the former theorems.In Chapter 2, we get a new Picard type theorem concerning omitted function, and prove the following result: Let f(z) be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k + 1, and all of whose poles are multiple, where k≥2 is an integer. Let the function a(z) = P(z)exp(Q(z)) (?) 0, where P and Q are polynomials such that (?)=∞. Then the function f^(k)(z) - a(z) has infinitely many zeros.In Chapter 3, we study a new kind of holomorphic functions, and get the following two results: (a). Let F be a family of functions holomorphic on a domain D (?) C. Let k≥2 be an integer and let h(z) be an holomorphic function on D, all of whose zeros have multiplicity at most k-1, such that h(z) has no common zeros with any f∈F. Assume also that the following two conditions hold for every f∈F(a) f(z) = 0 (?) f'(z)=h(z) and(b) f'(z) = h(z) (?) |f^(k)(z)|≤c, where c is a constant. Then F is normal on D.(b). Let F be a family of functions holomorphic on a domain D (?) C. Let k≠2 be an integer and let h(z) be an holomorphic function on D, all of zeros are simple, such that h(z) has no common zeros with any f∈F. Assume also 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) (?) f^(k+1)(z) = 0, where c is a constant. Then F is normal on D.In Chapter 4, at first, we obtain a criterion for quasinormal families, and proved: Let D (?) C be a simple connected domain, and {h_n} be a sequence of holomorphic functions on D, such that h_n (?) H'= cz^d on D, where H is holomorphic on D and H'≠0,∞, z∈D. Let {f_n} be a sequence of meromorphic functions on D, all of whose zeros have multiplicity at least k + 1, such that f_n^(k)(z)≠h_n(z) for all n and all z∈D, then {f_n} is quasinormal of order |d + 1| on D. Moreover, if no subsequence of [f_n] is normal at z₀∈D, then f_n^(k-1)(z) (?) H(z) - H(z₀) on D\{z₀}, and for all n, S(Δ(z₀,δ),f_n)≤k + 1. Then, we use this result to get a Picard type theorem, and proved: Let f be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k + 1, and let R (?) 0 be a rational function, then f^(k) - R has infinitely many zeros. This improved X.C.Pang, S.Nevo and L.Zalcman's result.In Chapter 5, we give some unknown problems.