f^(m)f^(k)的值分布及其相关的正规定则  

The Value Distribution of f^(m)f^(k)and its Related Normal Criteria

作　　者：李刘文 邹温林 LI Liuwen;ZOU Wenlin(Jinling Institute of Technology,Nanjing 211169,China;Nanjing Xiaozhuang University,Nanjing 211171,China)

机构地区：[1]金陵科技学院理学院,江苏南京211169 [2]南京晓庄学院信息工程学院,江苏南京211171

出　　处：《金陵科技学院学报》2024年第2期81-86,共6页Journal of Jinling Institute of Technology

基　　金：金陵科技学院高层次人才科研启动基金(jit-b-202043)。

摘　　要：应用值分布和正规族理论,对亚纯函数f(z)的m(≥3)次幂和其k阶导数的乘积f mf(k)的值分布进行讨论,得到了与其相关的一类正规定则,结果如下:设m(≥3),k为正整数,f(z)为复平面上的亚纯函数,且不为次数小于k的多项式,则3m-6/4 T(r,f)≤N(r,1/f^(m)f^((k))-1)+S(r,f)。设m(≥3),k为正整数,M为一正数,F是区域D内的一族亚纯函数,a(z)≠0为D内的全纯函数,如果对于F中的每个函数f,只有至少k级零点,则有:若f^(m)(z)f^(k)(z)=a(z)|f(k)(z)|<M,z∈D,则F在D内正规;若f^(m)(z)f^(k)(z)=a(z)|f(z)|>M,z∈D,则F在D内正规。By applying the theory of value distribution and normal families,this paper discusses the value distribution of the product f^(m)f^(k)of the m-th(m≥3)power of a meromorphic function f(z)and its k-th derivative,and obtains a class of normality criteria associated with it.The following results are obtained:Let m(≥3),k be any positive integer,and f(z)be a meromorphic function in the complex plane satisfying that it is not a polynomial with degree less than k,then 3m-64 T(r,f)≤N—r,1 f^(m)f^(k)-1+S(r,f).Let F be a family of meromorphic functions,and a(z)≠0 be a holomorphic function on the plane domain D.Let m(≥3),k be any positive integer,and M be a positive constant.Suppose that for each f∈F its zeros have multiplicity at least k,then F is normal on D,if f m(z)f(k)(z)=a(z)|f(k)(z)|<M,for z∈D,or f m(z)f(k)(z)=a(z)|f(z)|>M,for z∈D.

关 键 词：值分布 亚纯函数 零点 正规族 

分 类 号：O174.52[理学—数学]