scholarly journals Neo-Normal Functions in Arbitrary Regions

1972 ◽  
Vol 48 ◽  
pp. 19-36
Author(s):  
Kam-Fook Tse
Keyword(s):  
Unit Disk ◽  
Meromorphic Functions ◽  
Boundary Behavior ◽  
Universal Covering ◽  
Value Distribution ◽  
Covering Surface ◽  
Normal Functions ◽  
Definition Of ◽  
Simply Connected Region ◽  
Simply Connected

It is well known that many properties possessed by functions holomorphic and bounded in a region are also possessed by functions meromorphic and omitting three values. Noshiro [14] in 1938 and Lehto and Virtanen [12] in 1957 independently defined the notion of “normal functions” ; they and many others subsequently discovered that most properties concerning boundary behavior and value distribution acquired by meromorphic functions omitting three values in the unit disk (or more general, in a simply-connected region) are also valid properties of “normal functions” defined there. In their research on the problems of value distribution of normal functions, Lange [9], Gavrilov [5] and Gauthier [4] have discovered that functions normal in the disk are exactly those which omit three values “locally,” i.e., they do not possess any “p-sequence” (see above references). However, the definition of a function being normal in a region depends on the simply-connectedness of the region or its universal covering surface. It is thus difficult to judge if a function defined in an arbitrary region is normal.

Analytic Continuation of Power Series by Regular Generalized Weighted Means

1983 ◽  
Vol 26 (4) ◽  
pp. 454-463
Author(s):  
Bruce L. R. Shawyer ◽  
Ludwig Tomm
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Analytic Continuation ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Geometric Series ◽  
Summability Methods ◽  
Weighted Means ◽  
Simply Connected Region ◽  
Simply Connected

AbstractThe behaviour of summability transforms of power series outside their circles of convergence has been studied by many authors. In the case of the geometric series Luh [6] and Tomm [10] showed that there exist regular methods A which provide an analytic continuation into any given simply connected region G that contains the unit disc but not the point 1. Moreover, the Atransforms of the geometric series may be required to converge to any chosen analytic function on prescribed regions outside the unit circle. In this paper, these results are extended to power series representing other meromorphic functions. It is also shown that the summability methods involved may be chosen to be generalized weighted means previously introduced by Faulstich [1].

scholarly journals Vortices of the Mediterranean Sea: An Altimetric Perspective

2006 ◽  
Vol 36 (1) ◽  
pp. 87-103 ◽  
Author(s):  
Jordi Isern-Fontanet ◽  
Emilio García-Ladona ◽  
Jordi Font
Keyword(s):  
Mediterranean Sea ◽  
Mediterranean Basin ◽  
Common Definition ◽  
Coherent Vortices ◽  
The Mediterranean ◽  
Definition Of ◽  
Simply Connected Region ◽  
Simply Connected ◽  
First Time ◽  
The Mediterranean Basin

Abstract The presence of coherent vortices makes observed mesoscale fields of the ocean resemble two-dimensional turbulence. Using this analogy, a common definition of a coherent structure has been used to study the statistical properties of Mediterranean Sea vortices observed by satellite altimeters over a 7-yr period. A vortex has been defined as the simply connected region with values of the Okubo–Weiss parameter W < −0.2σW, where σW is the spatial standard deviation of W, and the same sign of vorticity. This definition is shown to be appropriate to detect and characterize, statistically, properties such as size, mean kinetic energy, and amplitude of vortices in the Mediterranean basin from sea level anomaly maps corresponding to the period from October 1992 to October 1999. The distribution of such properties for the Mediterranean vortices suggests a heuristic criterion to extract and select very coherent and long-lived vortices from the whole set of structures identified in altimetric maps. Such coherent vortices appear to be selected for amplitudes greater than 2σW, where the amplitude has been defined in terms of the Okubo–Weiss parameter rather than vorticity, and strongly correspond to those reported from observations with independent data. Systematic locating and tracking of such vortices provide, for the first time, a general picture of their preferential paths in the Mediterranean basin, which are characterized by complex but rather well defined patterns.

scholarly journals Boundary behavior of normal functions defined in the unit disk.

1963 ◽  
Vol 10 (1) ◽  
pp. 43-51 ◽  
Author(s):  
D. C. Rung
Keyword(s):  
Unit Disk ◽  
Boundary Behavior ◽  
Normal Functions

scholarly journals Some Results on Value Distribution of Meromorphic Functions in the Unit Disk

1969 ◽  
Vol 34 ◽  
pp. 105-119 ◽  
Author(s):  
Kam-Fook Tse
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Euclidean Distance ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Value Distribution ◽  
Open Unit ◽  
Open Unit Disk ◽  
Chordal Distance

Let C and D be the unit circle and the open unit disk respectively. We shall use p(z,z′) to represent the non-Euclidean distance [3, p. 263] between the two points z and z′ in D, and X(w, w′) to represent the chordal distance between the two points w and w′ on the Riemann Sphere Ω.

scholarly journals On Properties of Differences Polynomials about Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Jianming Qi ◽  
Jie Ding ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Special Class ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Class Difference ◽  
Difference Polynomial

We study the value distribution of a special class difference polynomial about finite order meromorphic function. Our methods of the proof are also different from ones in the previous results by Chen (2011), Liu and Laine (2010), and Liu and Yang (2009).

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

Stability of Thin Elastic Plates Covering an Arbitrary Simply Connected Region and Subject to any Admissible Boundary Conditions

1953 ◽  
Vol 20 (1) ◽  
pp. 23-29
Author(s):  
G. A. Zizicas
Keyword(s):  
Boundary Conditions ◽  
Direct Method ◽  
Elastic Plates ◽  
Simple Manner ◽  
Particular Solutions ◽  
Isotropic Plates ◽  
Simply Connected Region ◽  
Simply Connected ◽  
A Direct Method ◽  
Thin Elastic Plates

Abstract The Bergman method of solving boundary-value problems by means of particular solutions of the differential equation, which are constructed without reference to the boundary conditions, is applied to the problem of stability of thin elastic plates of an arbitrary simply connected shape and subject to any admissible boundary conditions. A direct method is presented for the construction of particular solutions that is applicable to both anisotropic and isotropic plates. Previous results of M. Z. Krzywoblocki for isotropic plates are obtained in a simple manner.

scholarly journals A Class of Nonlinear Integral Operators Preserving Double Subordinations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Oh Sang Kwon ◽  
Nak Eun Cho
Keyword(s):  
Unit Disk ◽  
Meromorphic Functions ◽  
Type Theorem ◽  
Integral Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Sandwich Type ◽  
Space Of Meromorphic Functions ◽  
Nonlinear Integral Operators

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain integral operators defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also considered.

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