On the angular boundary values of subharmonic functions in the unit disc

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
S. L. Berberyan ◽  
P. M. Gauthier
Keyword(s):  
Unit Disc ◽  
Boundary Values ◽  
Subharmonic Functions ◽  
Angular Boundary

scholarly journals Riesz-Martin representation for positive super-polyharmonic functions in A Riemannian manifold

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
V. Anandam ◽  
S. I. Othman
Keyword(s):  
Riemannian Manifold ◽  
Complex Plane ◽  
Unit Disc ◽  
Biharmonic Function ◽  
Subharmonic Functions ◽  
Polyharmonic Functions ◽  
Superharmonic Functions ◽  
Martin Representation

Letube a super-biharmonic function, that is,Δ2u≥0, on the unit discDin the complex plane, satisfying certain conditions. Then it has been shown thatuhas a representation analogous to the Poisson-Jensen representation for subharmonic functions onD. In the same vein, it is shown here that a functionuon any Green domainΩin a Riemannian manifold satisfying the conditions(−Δ)iu≥0for0≤i≤mhas a representation analogous to the Riesz-Martin representation for positive superharmonic functions onΩ.

Ideals in Rings of Analytic Functions with Smooth Boundary Values

1970 ◽  
Vol 22 (6) ◽  
pp. 1266-1283 ◽  
Author(s):  
B. A. Taylor ◽  
D. L. Williams
Keyword(s):  
Banach Algebra ◽  
Unit Disc ◽  
Smooth Boundary ◽  
Topological Algebra ◽  
Boundary Function ◽  
Open Unit ◽  
Boundary Values ◽  
Open Unit Disc ◽  
Algebra Of Functions ◽  
Derivatives Of

LetAdenote the Banach algebra of functions analytic in the open unit discDand continuous in. Iffand its firstmderivatives belong toA,then the boundary functionf(eiθ)belongs toCm(∂D). The spaceAmof all such functions is a Banach algebra with the topology induced byCm(∂D).If all the derivatives of/ belong toA,then the boundary function belongs toC∞(∂D), and the spaceA∞all such functions is a topological algebra with the topology induced byC∞(∂D). In this paper we determine the structure of the closed ideals ofA∞(Theorem 5.3).Beurling and Rudin (see e.g. [7, pp. 82-89;10]) have characterized the closed ideals ofA, and their solution suggests a possible structure for the closed ideals ofA∞.

Interpolation by Linear Sums of Harmonic Measures

1975 ◽  
Vol 18 (2) ◽  
pp. 249-253
Author(s):  
Marvin Ortel
Keyword(s):  
Unit Circle ◽  
Harmonic Measure ◽  
Unit Disc ◽  
Boundary Values ◽  
Harmonic Measures ◽  
Image Position ◽  
Interior Points

Let α be an open arc on the unit circleand for z=reiθ, 0 ≤ r < 1, let(1)The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.

scholarly journals Tangential Boundary Properties of Arbitrary Functions in the Unit Disc

1972 ◽  
Vol 46 ◽  
pp. 111-120 ◽  
Author(s):  
Hidenobu Yoshida
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Boundary Properties ◽  
Angular Boundary ◽  
Boundary Behaviors

1. By the method of Dolzhenko’s paper, we studied relations between non-tangetial (angular) boundary behaviors and horocyclic boundary behaviors of arbitrary functions defined in the open unit disc of the complex plane in [8]. Vessey [5], [6] investigated the behavior of arbitrary functions on paths which are “more tangential” than horocycles. The purpose of the present paper is to prove the fact that is sharper than the results in Vessey [5], [6], and generalize the results in [8] to obtain the connection between behaviors on two “more tangential” angles.

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