亚纯函数的一个正规定则

We proved:Let F be a family of meromorphic functions in a domain D and a≠0,b∈C.If f′(z)-a(f(z))~2≠b,f≠0 and the poles of f(z)are of multiplicity＞=3 for each f(z)∈F,then F is normal in D.     

全纯函数族的正规定则

Let f be a holomorphic function on a domain D Lontaiu in C, and let a be a finite complex number. We denote by -↑Ef/(a) = {z ∈ D : f(z) = a, ignoring multiplicity} the set of all distinct α-points of f. Let F be a family of holomorphic functions on D. If there exist three finite values a, b(≠ O,α) and c(≠ O) such that for every f ∈F, -↑Ef′/(0) Lontain in -↑Ef(α） and -↑Ef′ (b) Lontain in -↑Ef(c), then F is a normal family on D.     

与分担全纯函数相关的正规族

设F是在区域D内的一族亚纯函数,其零点重级至少为k,k是一个正整数,a(z)(≠0)在区域D内全纯.若对于任意的f∈F,有(1)f(z)与a(z)没有公共的零点;(2)f(z)=0f(k)(z)=a(z)■0|f~((k+1))(z)-a'(x)||a(z)|,则F在D内正规.     

An Extension of an Inversion Theorem of Carlitz

For q≠1,Gauss q-coefficients may be defined as followswhere0≤k≤n and(q)_0=1.For k>n one may define(?)=0.Theorem Let{a_j(z)}and{b_j(z)}be any two sequences of complex valuedfunctions of z and let q(q≠1)be an arbitrary complex number such that     

涉及分担函数的正规定则

设k为正整数,M为正数;F为区域D内的亚纯函数族,且其零点重级至少为k;h为D内的亚纯函数(h(z)≠0,∞),且h(z)的极点重级至多为k.若对任意给定的函数f∈F,f与f~((k))分担0,且f~((k))(z)-h(z)=0?|f(z)|≥M,则F在D内正规.     

涉及微分多项式的正规定则（英文）

本文获得如下结果：设φ（z)为区域G内一不恒为零的亚纯函数，a1(z),a2(z),．…，ak(z)为区域G内的全纯函数，F＝{f}为G内一亚纯函数族，若对每一f∈F，在G内恒有f(z)，f(z)≠0,f^(k)(z) a1(z)f^(f-1)(z) … ak(z)f(z)≠φ（z),且与φ（z)没有公共极点，则F在G内正规。     

DISTRIBUTION OF THE（0，∞）ACCUMULATIVE LINES OF MEROMORPHIC FUNCTIONS

Suppose that f(z)is a meromorphic function of order λ（0<λ<+∞）and of lower order μ in the plane.Let ρ be a positive number such that μ≤ρ≤λ.(1)If f^(l)(z)(0≤l<+∞）has p(1≤p<+∞）finite nonzero deficient valnes αi(i=1,…，p)with deficiencies δ（αi,f^(l)),then f(z)has a (0,∞）accumulative line of order ≥ρin any angular domain whose vertex is at the origin and whose magnitude is larger than max(π/ρ，2π-4/ρ ∑i=1^p arcsin √δ（αi,f^(l))/2).(2)If f(z) has only p(0<p<+∞）（0,∞）,accumulative lines of order≥ρ：arg z=θk(0≤θ1<θ2<…<θp<2π，θp+1=θ1+2π）,then λ≤π/ω，where ω=min I≤k≤p(θk+1-θk),provided that f^(l)(z)(0≤l<+∞）has a finite nonzero deficient value.     

亚纯函数的分担值与正规族

设k(≥2)为正整数,M为一个正数,h(z)为区域D内的一个全纯函数,h≠0,F为区域D内的一族亚纯函数,其中每个函数的零点重级至少为k+1,极点重级至少为2.若任意f∈F,f~((k))(z)=h(z)|f(z)|≥M,则F在D内正规.     

On a Theorem of Milin

Let S be class of functions f(z)=z a_2 z~2 … analytic and univalent in the unit disk D, and let 1.M.Milin proved that Theorem A. If f∈S and then where d_o(h)=1and In fact,the result is deduced from Milan′s Tauberian Theorem. Here     

具有指定亏函数和指定亏量的素函数

设F(z)为亚纯函数,若F可表为 F(z)=f°g(z) (1) 其中g为整函数,f为亚纯函数(当f为有理函数时,g可为亚纯函数)。我们称(1)为F的一个分解。若F的任何形如(1)的分解都只能是f或g为线性函数,则称F为素的。如果F的任何形如(1)的分解都只能以g为多项式或f为有理函数的形式出现,则称F为拟素的。