Isovolumetric inequalities for the least harmonic majorant of |x|p

1988 ◽  
pp. 240-241
Author(s):  
Makoto Sakai
Keyword(s):  
Harmonic Majorant

scholarly journals A note on n-harmonic majorants

1986 ◽  
Vol 34 (3) ◽  
pp. 461-472
Author(s):  
Hong Oh Kim ◽  
Chang Ock Lee
Keyword(s):  
Harmonic Function ◽  
Complex Plane ◽  
Unit Disc ◽  
Dirichlet Integral ◽  
Harmonic Majorant ◽  
Harmonic Majorants

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

Extreme Points in H1(R)

1967 ◽  
Vol 19 ◽  
pp. 312-320 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Set ◽  
Complete Solution ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Open Riemann Surface ◽  
Harmonic Majorant ◽  
Closed Convex Set

Let R be an open Riemann surface. ƒ belongs to H1(R) if ƒ is holomorphic on R and if the subharmonic function |ƒ| has a harmonie majorant on R. Let p be in R and define ||ƒ|| to be the value at p of the least harmonic majorant of |ƒ|. ||ƒ|| is a norm on the linear space H1(R), and with this norm H1(R) is a Banach space (7). The unit ball of H1(R) is the closed convex set of all ƒ in H1(R) with ||ƒ|| ⩽ 1. Problem: What are the extreme points of the unit ball of H1(R)? de Leeuw and Rudin have given a complete solution to this problem where R is the open unit disk (1).

scholarly journals On Level Curves of Harmonic and Analytic Functions on Riemann Surfaces

1969 ◽  
Vol 34 ◽  
pp. 77-87
Author(s):  
Shinji Yamashitad
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Vector Lattice ◽  
Level Curves ◽  
Harmonic Majorant ◽  
Harmonic And Analytic Functions ◽  
Hyperbolic Riemann Surface

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

Theorems of the Phragmén–Lindelöf Type for Subfunctions in a Cone

2016 ◽  
Vol 60 (3) ◽  
pp. 739-751 ◽  
Author(s):  
Lei Qiao ◽  
Guoshuang Pan
Keyword(s):  
Maximum Principle ◽  
Unbounded Domain ◽  
Type Theorem ◽  
The Maximum Principle ◽  
Lindelöf Theorem ◽  
Harmonic Majorant

AbstractOur first aim in this paper is to deal with the maximum principle for subfunctions in an arbitrary unbounded domain. As an application, we next give a result concerning the classical Phragmén–Lindelöf theorem for subfunctions in a cone. For a subfunction defined in a cone that is dominated on the boundary by a certain function, we finally generalize the Phragmén–Lindelöf type theorem by making a generalized harmonic majorant of it.

scholarly journals RETRACTED ARTICLE: A new application of boundary integral behaviors of harmonic functions to the least harmonic majorant

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Minghua Han ◽  
Jianguo Sun ◽  
Gaoying Xue
Keyword(s):  
Subharmonic Function ◽  
Harmonic Functions ◽  
Boundary Integral ◽  
Retracted Article ◽  
New Type ◽  
Harmonic Majorant

Abstract Our main aim in this paper is to obtain a new type of boundary integral behaviors of harmonic functions in a smooth cone. As an application, the least harmonic majorant of a nonnegative subharmonic function is also given.

scholarly journals On Some Families of Analytic Functions on Riemann Surfaces

1968 ◽  
Vol 31 ◽  
pp. 57-68 ◽  
Author(s):  
Shinji Yamashita
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Harmonic Majorant

Throughout this paper all functions are single-valued. Let R be a Riemann surface. We shall denote by φ∧ the least harmonic majorant of a function φ defined in R if it has the meaning.

scholarly journals A THEOREM OF PHRAGMÉN-LINDELÖF TYPE FOR SUBFUNCTIONS IN A CONE

2011 ◽  
Vol 53 (3) ◽  
pp. 599-610 ◽  
Author(s):  
LEI QIAO ◽  
GUANTIE DENG
Keyword(s):  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Stationary Schrödinger Operator ◽  
Lindelöf Theorem ◽  
Harmonic Majorant

AbstractFor a subfunction u, associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we generalise the classical Phragmén-Lindelöf theorem by making an a-harmonic majorant of u.

On a Class of Analytic Functions of Smirnov

1979 ◽  
Vol 31 (1) ◽  
pp. 181-183
Author(s):  
J. L. Schiff
Keyword(s):  
Analytic Functions ◽  
Logarithmic Capacity ◽  
Compact Set ◽  
Smirnov Class ◽  
Capacity Zero ◽  
Planar Surfaces ◽  
Important Development ◽  
Harmonic Majorant ◽  
Totally Disconnected

The class S of functions under study in this paper was introduced by V. I. Smirnov in 1932. This class was subsequently investigated by various authors, a pertinent paper to the present wrork being that of Tumarkin and Havinson [2], who showed that a plane compact set of logarithmic capacity zero is 5-removable. Another important development, due to Yamashita [3], wras that the class 5 could be characterized as those analytic functions ƒ for which log+ |ƒ| has a quasi-bounded harmonic majorant.In what follows, we discuss the Smirnov class in the context of planar surfaces, exploiting some ideas in the work of Hejhal [1] to establish that a closed, bounded, totally disconnected set is S-removable if and only if its complement belongs to the null class Os.

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