与fL(f)相关的正规定则

Normal criteria related to fL(f)

设F是区域D内的一亚纯函数族,k是任一正整数,M为一正数,对于F中的每个函数f,只有至少t级零点,当1 k 4时,t=k+1,当k 5时,t=k.置L(f)(z)=f(k)(z)+ak-1(z)f(k-1)(z)+…+a1(z)f′(z)+a0(z)f(z)+b(z),a(z)≠0,a0(z),a1(z),…,ak-1(z),b(z)为D内的全纯函数,则有(i)若f(z)L(f)(z)=a(z)|L(f)(z)|<M,z∈D,则F在D内正规.(ii)若f(z)L(f)(z)=a(z)|f(z)|>M,z∈D,则F在D内正规.

Let F be a family of functions meromorphic on the plane domain D. Let k be a positive integer and M be a positive constant. Suppose that for each f in F its zeros have multiplicity at least t, when 1≤k≤4, t = k ＋ 1 and when k≥5,t=k. Set L（f）（z）=f^（k）（z）＋ak-1（z）^（k-1）（z）＋…＋a1（z）f（z）＋a0（z）f（z）＋b（z）, where a（z）≠0,a0（z）,a1（z）,…,ak-1（z）,b（z） be holomorphic functions on D. Then F is normal on D if f（z）L （f）（z） = a（z）=〉｜L（f）（z）｜〈M for z in U or f（z）L（f）（z） = a（z）=〉｜f（z）｜〉M for z in D.