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 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.

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

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 Analytic Functions on Some Riemann Surfaces, II

1963 ◽  
Vol 23 ◽  
pp. 153-164 ◽  
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Harmonic Measure ◽  
Analytic Functions ◽  
Boundary Point ◽  
Riemann Surfaces ◽  
Positive Answer ◽  
Open Riemann Surface ◽  
Hyperbolic Riemann Surface

In their paper [12], Toda and the author have concerned themselves in the followingTheorem of Kuramochi. Let R be a hyperbolic Riemann surface of the class OHB(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 OAB(OAD, resp.) (Kuramochi [4]).They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.

scholarly journals P-harmonic dimensions on ends

1993 ◽  
Vol 132 ◽  
pp. 131-139
Author(s):  
Michihiko Kawamura ◽  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Boundary Component ◽  
Harmonic Functions ◽  
Ideal Boundary ◽  
Boundary Values ◽  
Hölder Continuous ◽  
Closed Curves ◽  
Open Riemann Surface ◽  
Relative Boundary ◽  
Bounded Harmonic Functions

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

scholarly journals On global cluster sets for functions meromorphic on some Riemann surfaces

1978 ◽  
Vol 70 ◽  
pp. 1-6 ◽  
Author(s):  
Shigeo Segawa
Keyword(s):  
Riemann Surfaces ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
A Value ◽  
Cluster Set ◽  
Global Cluster ◽  
Cluster Sets ◽  
Extended Complex ◽  
Definition Of ◽  
Present Setting

Consider a single-valued meromorphic function w = f(p) defined on an open Riemann surface R with an ideal boundary β. In [1], Collingwood and Cartwright introduced the global cluster set for a function meromorphic on the unit disk. Generalizing the definition of global cluster sets to our present setting, we define the global cluster set for w = f(p) as follows;A value w in the extended complex plane is called a cluster value at β if there exists a sequence in R converging to β such that

scholarly journals Banach spaces of bounded solutions of Δu = Pu (P ≥ 0) on hyperbolic riemann surfaces

1974 ◽  
Vol 53 ◽  
pp. 141-155 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Banach Spaces ◽  
Riemann Surfaces ◽  
Space Structure ◽  
Bounded Solutions ◽  
Linear Isometry ◽  
Hölder Continuous ◽  
Hyperbolic Riemann Surfaces ◽  
Hyperbolic Riemann Surface

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

Sign in / Sign up

Export Citation Format

Share Document