On Normality Of Meromorphic Functions

This paper consisted of two parts which discussed two topics respectively.One topic is about theory of normal family,and the other is aboutshared value and periodic function.The first part of paper start from a problem about normal family andomitted function raised by Fang Mingliang,and it extended the work byZalcman in this direction.A mainly theorem we got in this part is:Letk≥2 be an integer and (?)_a family of functions meromorphic on a domainD in C,all of whose poles are multiple and whose zeros all have multiplicityat least k+1.Let h be a function meromorphic on domain D,h(?)0,∞.Suppose that for each f∈(?),f^k(z)≠h(z) for z∈D.Then (?) is a normalfamily on D.We raised a problem about shared value and periodic function in thesecond part:Suppose / be a transcendental meromorphic function satisfiedf(z)∈E(?)f(z+1)∈E,where E is a set consisted of three distinctfinite complex numbers.Then whether it be right or not that f must bea periodic function?In this direction we discussed some cases,and got thefollowing theorem with additional conditions:Let f be a transcendentalentire function with finite order,f(z) and f(z+1) shared two finite complexnumbers counting multiplicity.Then f(z) must be periodic function withperiod 1.In this part we also do some simple discussing with a result ofJ.Langley.