涉及分担值的亚纯函数的正规性

Normal families and shared values of meromorphic functions

研究把前人所作定理条件中的f(z)换成fn(z)看结论是否仍然成立.采用Zalcman引理和正规族的相关结论以及Nevanlinna第一、二基本定理等方法,研究与分担值相关的亚纯函数的正规性问题,得到一个新的结论.设F是区域D内的一族亚纯函数,k,n≥3是正整数,a,c是2个非零有穷复数,b,d是正实数,若f(z)∈F,f的零点重数至少是k,若fn(z)f(k)(z)=a>|f(k)(z)|≤b,f(k)(z)=c>|fn(z)f(k)(z)|≥d,则F在D内正规.

In this paper,major research makes the previous theorem f（z） replaced fn（z）,conclusions are still valid.By using Zalcman′s lemma and the formal conclusions of normal families and the first and second Nevanlinna fundamental theorem,normal families of meromorphic functions concerning sharing values was studied.The following theorem is got that let F be a family of meromorphic functions in domain D,all of whose zeros are of multiplicity at least k,k and n（n ≥3） are two positive integers,a and c be two nonzero finite complex numbers,b,d be two positive real numbers,if f（z）∈F,fn（z）f（k）（z）=a｜f（k）（z）｜≤b,f（k）（z）=c｜fn（z）f（k）（z）｜≥d,then F is normal in domain D.