The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D C C, all of whose zeros have multiplicity at least k, where k 〉 2 is an integer. And let h（z）≠ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈F： （a） f（z） = 0 [f^（k）（z）｜ 〈 ｜h（z）｜; （b） f^（k）（z）≠ h（z）. Then F is normal on
In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved： Let F and G be two families of functions meromorphic on a domain D C C, a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f 9~, there exists g C G such that f（z） and g（z） share the values a1, a2, a3, a4, then F is normal on D.
We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜α 〉 C for every z ∈ D is normal in D.For α = 1,the same assumptions imply quasi-normality but not necessarily normality.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of functions holomorphic on a domain D C, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. Let h（z） ≠ 0 and oo be a meromorphic function on D. Assume that the following two conditions hold for every f C Dr ： （a） f（z） = 0 =→ ｜f（k）（z）｜ 〈｜h（z）｜. （b） f（k）（z） ≠ h（z）. Then F is normal on D.
In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.
Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. 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.If fn≠hn, then {fn} is normal on D.
