Theory of analytic positive and single-sheeted mappings by means of generalized entire and meromorphic functions with a finite number of zeros and poles

1970 ◽  
Vol 22 (1) ◽  
pp. 104-110
Author(s):  
P. G. Todorov
Keyword(s):  
Finite Number ◽  
Meromorphic Functions ◽  
Number Of Zeros ◽  
Zeros And Poles

Two semigroup rings associated to a finite set of meromorphic functions

2020 ◽  
Vol 70 (5) ◽  
pp. 1249-1257
Author(s):  
Mircea Cimpoeaş
Keyword(s):  
Finite Number ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Semigroup Rings ◽  
General Properties ◽  
Number Of Zeros ◽  
Finite Set ◽  
Zeros And Poles

AbstractWe fix z0 ∈ ℂ and a field 𝔽 with ℂ ⊂ 𝔽 ⊂ 𝓜z0 := the field of germs of meromorphic functions at z0. We fix f1, …, fr ∈ 𝓜z0 and we consider the 𝔽-algebras S := 𝔽[f1, …, fr] and $\begin{array}{} \overline S: = \mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]. \end{array} $ We present the general properties of the semigroup rings$$\begin{array}{} \displaystyle S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\\overline S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{array} $$and we tackle in detail the case 𝔽 = 𝓜<1, the field of meromorphic functions of order < 1, and fj’s are meromorphic functions over ℂ of finite order with a finite number of zeros and poles.

On the Topological Theory of Functions

1951 ◽  
Vol 3 ◽  
pp. 276-289 ◽  
Author(s):  
James A. Jenkins
Keyword(s):  
Finite Number ◽  
Meromorphic Functions ◽  
Parabolic Type ◽  
Hyperbolic Type ◽  
Open Disc ◽  
Topological Theory ◽  
Number Of Zeros ◽  
Simply Connected Domain ◽  
The Subject ◽  
Simply Connected

The present paper constitutes a continuation of the ideas and methods of M. Morse and M. Heins [1]. As in that work the subject treated is the theory of deformation classes of meromorphic functions and interior transformations. There the functions considered were defined over the open disc &lt; 1 and had only a finite number of zeros, poles and branch point antecedents. It is possible to transfer the results obtained to the situation where the domain of definition is any simply-connected domain of hyperbolic type or, alternatively, of parabolic type.

LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

2021 ◽  
Vol 9 (1) ◽  
pp. 164-170
Author(s):  
Y. Gal ◽  
M. Zabolotskyi ◽  
M. Mostova
Keyword(s):  
Blaschke Product ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Counting Function ◽  
Finite System ◽  
Blaschke Products ◽  
Nevanlinna Characteristic ◽  
Number Of Zeros ◽  
Accumulation Points ◽  
Theory Of Value

The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

scholarly journals On the Ritt order and type of a certain class of functions defined byBE-Dirichletian elements

1999 ◽  
Vol 22 (3) ◽  
pp. 445-458
Author(s):  
Marcel Berland
Keyword(s):  
Finite Number ◽  
Entire Functions ◽  
Number Of Zeros ◽  
Order And Type ◽  
Class Of Functions

We introduce the notions of Ritt order and type to functions defined by the series∑n=1∞fn(σ+iτ0)exp(−sλn),      s=σ+iτ,(σ,τ)∈R×R                                            (*)indexed byτ0onR, where(λn)1∞is aD-sequence and(fn)1∞is a sequence of entire functions of bounded index with at most a finite number of zeros. By definition, the series areBE-Dirichletian elements. The notions of order and type of functions, defined byB-Dirichletian elements, are considered in [3, 4]. In this paper, using a technique similar to that used by M. Blambert and M. Berland [6], we prove the same properties of Ritt order and type for these functions.

Escaping points of meromorphic functions with a finite number of poles

2005 ◽  
Vol 96 (1) ◽  
pp. 225-245 ◽  
Author(s):  
P. J. Rippon ◽  
G. M. Stallard
Keyword(s):  
Finite Number ◽  
Meromorphic Functions
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