无穷级整函数的一个缺项定理

设f(z)=sum from n=0 to∞a_nz~n为整函数,为了显示它的缺项,我们把它表示成f(z/)=sum from n-1 to∞a(_λ_n)z(~λ_n)1929年,G.Polya猜测:当整函数(1)为有穷级时,若其残存指数序列{λ_n}满足Fabry缺项条件(?)λ_n/n=∞,则(?)(In L(r,f)/In M(r,f))=1成立其中M(r,f)=(?)|f(z)|,L(r,f)=(?)|f(z)|.1963年,Fuchs证实了这个猜测.当整函数(1)为无穷级时,T.Kovari和谢晖春分别在加强的缺项条件     

一类整函数的亏函数

1964年,庄圻泰教授在整函数的情况彻底解决了R.Nevanlinna提出的关于第二基本定理的推广问题,从而为亏函数的研究开创了道路。在文[2]中,我们用庄圻泰引进的Wronskin行列式作为工具解决了F.Nevanlinna猜想的广泛形式。本文继续用这一工具从推广文[3]中定理2出发,继而给出了一类整函数的亏函数个数。最后对庄圻泰提出的一个问题给出了肯定的回答。     

关于整函数的亏函数

设ｆ（ζ）为超越整函数，则它和它的各级导函数与原函数的亏函数的亏量满足关系这里的集合Ａｆ，将在§１中给出．设ｆ（ｚ）的下级μ＜＋∞．如果上式成立等号，则有ｉ）ｆ（ｚ）的级与下级相等，且为正整数：ｉｉ）Ａｆ中的元素个数不超过μ；ｉｉｉ）每个δ（ａｋ，ι，ｆ）均为的整数倍；ｉｖ）每个ａｋ，ι均为ｆ（ｋ）的渐近函数．     

缺项广义Dirichlet级数所表示的整函数的增长性

对于有限级广义Dirichlet级数所表示的整函数f(s),引进级ρ_R,型■_R,分别讨论它们与在水平直线上的级ρ_L,型■_L之间的关系．应用于,Taylor级数时,有类似结果．     

一类整函数系数线性微分方程解的增长级和零点

在本文中假设微分方程的系数为有限级整函数且满足：对于每个不恒等于零的系数Aj(j为整数且0≤j≤k-1)，其零点收敛指数小于其增长级，且当Ai≠0,Aj≠0(i≠j),Ai/Aj的增长级等于Ai与Aj增长级的最大值，以及自由项F为有限级整函数。我们研究了线性微分方程f^(k) Ak-1f(^k-1) … A0f=f,k≥2解的增长级和零点收敛指数，得到在一定条件下，方程解的增长级及零点收敛指数的精确估计。     

整代数体函数与其导函数的增长级

对整代数体函数与其导函数的增长性进行了研究,得到了整代数体函数与其导函数的特征函数之间关系式,并证明了它们具有相同的级和下级,从而将亚纯函数的相应结果推广到了整代数体函数.     

Laplace-Stieltjes 变换所定义的无限级整函数的增长性

该文系统地研究了在全平面上收敛的无限级Laplace-Stieltjes变换的增长性,得到了两个充要条件,推广了全平面上Dirichlet级数的有关结果.     

Laplace-Stieltjes变换所确定的整函数的对数级

研究了全平面上收敛的零级Laplce-Stieltjes变换的增长性问题,通过定义对数级和对数下级,得到了零级Laplace-Stieltjes变换具有对数级和对数下级的特征性质,推广了Dirichlet级数相关结果.     

关于整函数的零点分布与分担值定理

本文主要讨论整函数零点分布与分担值定理的联系，并运用新颖的方法证明几个有趣的定理。     

有限级超越整函数的差分多项式的值分布

研究了差分多项式H(z)=POk∑(i=1)a_if(z+c_i)的值分布,其中f是有限级超越整函数,P(f)是,的多项式,κ≥2,ci(i=1,…,k)是互不相同的常数,α_i(i=1,…,κ)是非零常数.得到了H(z)-a和H(z)-α(z)的零点的个数的估计,其中a∈C且α(z)(■0)为小函数.讨论了H(z)的非零有限Borel例外值的不存在性.     

On some bidimensional denumerable chains of infinite order

We study homogeneous chains of infinite order (ξ_t)_{t∈$\text{Z}$} with the set of states taken to be $\text{X=(}\text{N}\text{{0,1})x{-1,1}}$. Our approach is to interpret the half-infinite sequence ..., ξ_?n,..., ξ_?1, ξ₀, where $\text{ξ}{}_{\mathrm{t}}\text{=(i}{}_{\mathrm{t}}\text{, ε}{}_{\mathrm{t}}\text{) ∈ X, t ∈}\text{Z}$ as the continued fraction to the nearer integer expansion (read inversely) of a y ? [?$\text{1}\text{2}$,$\text{1}\text{2}$]. Thus, we are led to study certain Y-valued Markov chains, where Y = [?$\text{1}\text{2}$, $\text{1}\text{2}$] and then by making use of their properties we establish the existence of denumerable chains of infinite order under conditions different from those given in Theorem 2.3.8 of Iosifescu-Theodorescu (1969). A (weak) variant of mixing is proved as well.     

On the growth of meromorphic functions of infinite order

Letf be a meromorphic function of infinite order,T(r, f) its Nevanlinna (or Ahlfors-Shimizu) characteristic, andM(r, f) its maximum modulus. It is proved that $$\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{rT'(r,f)}} \leqslant \pi and\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{T(r,f)\psi (log T(r,f))}} = 0$$ . if ? (x)/x is non-decreasing, ?′(x)<-√?(x) and ∝^∞ dx/?(x) < ∞.     

On sums of roots of infinite order entire functions

A class of infinite order entire functions is considered. Estimates for sums of the roots are derived. These estimates supplement the Hadamard theorem. Moreover, we establish a new estimate for the counting function of the roots, which in appropriate situations can be more useful than the Jensen inequality.     

On compact subsets in coechelon spaces of infinite order

For coechelon spaces of infinite order it is proved that every compact subset of is contained in a closed absolutely convex hull of some null sequence if and only if the matrix is regularly decreasing.

     

On the Mean Values of the Dirichlet Series

For Dirichlet series with arbitrary abscissa of absolute convergence, we investigate the relationhip between the increase in the maximum term and , q ∈ (0,+∞).__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1501–1512, November, 2004.