scholarly journals Some Theorems on Open Riemann Surfaces

1951 ◽  
Vol 3 ◽  
pp. 141-145 ◽  
Author(s):  
Masatsugu Tsuji
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Green’S Function ◽  
Green's Function ◽  
Riemann Surfaces ◽  
Dirichlet Integral ◽  
Open Riemann Surface ◽  
Surface Spread

Let F be an open Riemann surface spread over the z-plane. We say that F is of positive or null boundary, according as there exists a Green’s function on F or not, Let u(z) be a harmonic function on Fand be its Dirichlet integral As R. Nevanlinna proved, if F is of null boundary, there exists no one-valued non-constant harmonic function on F5 whose Dirichlet integral is finite, This Nevanlinna’s theorem was proved very simply by Kuroda.

Range Sets And Bmo Norms of Analytic Functions

1984 ◽  
Vol 36 (4) ◽  
pp. 747-755 ◽  
Author(s):  
Shoji Kobayashi
Keyword(s):  
Riemann Surface ◽  
Analytic Function ◽  
Green's Function ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Bounded Mean Oscillation ◽  
Open Riemann Surface ◽  
Mean Oscillation ◽  
Image Position ◽  
Dirichlet Integrals

In this paper we are concerned with the space BMOA of analytic functions of bounded mean oscillation for Riemann surfaces, and it is shown that for any analytic function on a Riemann surface the area of its range set bounds the square of its BMO norm, from which it is seen as an immediate corollary that the space BMOA includes the space AD of analytic functions with finite Dirichlet integrals.Let R be an open Riemann surface which possesses a Green's function, i.e., R ∉ OG, and f b e an analytic function defined on R. The Dirichletintegral DR(f) = D(f) of f on R is defined by1.1and we denote by AD(R) the space of all functions f analytic on R for which D(f) < +∞.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

scholarly journals Open Riemann Surface with Null Boundary

1951 ◽  
Vol 3 ◽  
pp. 73-79 ◽  
Author(s):  
Kiyoshi Noshiro
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Capacity Zero

Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.

On a Uniqueness Theorem

1979 ◽  
Vol 31 (5) ◽  
pp. 1072-1076
Author(s):  
Mikio Niimura
Keyword(s):  
Riemann Surface ◽  
Uniqueness Theorem ◽  
Analytic Functions ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Uniqueness Theorems ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
Hyperbolic Riemann Surfaces

The classical uniqueness theorems of Riesz and Koebe show an important characteristic of boundary behavior of analytic functions in the unit disc. The purpose of this note is to discuss these uniqueness theorems on hyperbolic Riemann surfaces. It will be necessary to give additional hypotheses because Riemann surfaces can be very nasty. So, in this note the Wiener compactification will be used as ideal boundary of Riemann surfaces. The Theorem, Corollaries 1, 2 and 3 are of Riesz type, Riesz-Nevanlinna type, Koebe type and Koebe-Nevanlinna type respectively. Corollaries 4 and 5 are general forms of Corollaries 2 and 3 respectively.Let f be a mapping from an open Riemann surface R into a Riemann surface R′.

scholarly journals Area growth and Green's function of Riemann surfaces

1992 ◽  
Vol 30 (1-2) ◽  
pp. 83-92 ◽  
Author(s):  
J. L. Fernández ◽  
J. M. Rodríguez
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Riemann Surfaces ◽  
Area Growth

scholarly journals Strong disk property for domains in open Riemann surfaces

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1711-1716
Author(s):  
Makoto Abe ◽  
Gou Nakamura
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Approximation Property ◽  
Stein Space ◽  
Open Riemann Surface ◽  
Holomorphic Approximation ◽  
Open Set ◽  
Pure Dimension

We study the relation between the holomorphic approximation property and the strong disk property for an open set of an open Riemann surface or a Stein space of pure dimension 1.

scholarly journals On Principal Function Problem

1970 ◽  
Vol 38 ◽  
pp. 85-90 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Boundary Behavior ◽  
Ideal Boundary ◽  
Principal Function ◽  
Open Riemann Surface ◽  
The Given ◽  
The Ideal

Sario’s theory of principal functions fully discussed in his research monograph [3] with Rodin stems from the principal function problem which is to find a harmonic function p on an open Riemann surface R imitating the ideal boundary behavior of the given harmonic function s in a neighborhood A of the ideal boundary δ of R.

scholarly journals Functions meromorphic on some Riemann surfaces

1978 ◽  
Vol 70 ◽  
pp. 41-45
Author(s):  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
A Value ◽  
Extended Complex ◽  
The Ideal ◽  
Extended Complex Plane

Consider an open Riemann surface R and a single-valued meromorphic function w = f(p) defined on R. A value w0 in the extended complex plane is said to be a cluster value for w = f(p) if there exists a sequence {pn } in R accumulating at the ideal boundary of R such that

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