scholarly journals On the derivatives at the origin of entire harmonic functions

1979 ◽  
Vol 20 (2) ◽  
pp. 147-154 ◽  
Author(s):  
D. H. Armitage
Keyword(s):  
Entire Function ◽  
Euclidean Space ◽  
Complex Plane ◽  
Harmonic Functions ◽  
Image Position ◽  
Derivatives Of

If f is an entire function in the complex plane such thatwhere 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).

scholarly journals Integral Harnack inequality

1985 ◽  
Vol 26 (2) ◽  
pp. 115-120 ◽  
Author(s):  
Murali Rao
Keyword(s):  
Euclidean Space ◽  
Hausdorff Measure ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Convex Domains ◽  
Dimensional Hausdorff Measure ◽  
Image Position ◽  
Continuous Boundary ◽  
Integral Version

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

Some Imbedding Theorems for Sobolev Spaces

1971 ◽  
Vol 23 (3) ◽  
pp. 517-530 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Dimensional Euclidean Space ◽  
Dense Subspace ◽  
Bounded Subset ◽  
Partial Derivatives ◽  
Image Position ◽  
Derivatives Of

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

scholarly journals Delange's characterization of the sine function

1974 ◽  
Vol 15 (1) ◽  
pp. 66-68 ◽  
Author(s):  
Chin-Hung Ching ◽  
Charles K. Chui
Keyword(s):  
Entire Function ◽  
Analytic Continuation ◽  
Complex Plane ◽  
Real Line ◽  
Sine Function ◽  
The Real ◽  
Image Position

In [2], H. Delange gives the following characterization of the sine function.Theorem A. f(x)=sin x is the only infinitely differentiable real-valued function on the real line such that f'(O)= 1 andfor all real x and n = 0,1,2,….It is clear that, if f satisfies (1), then the analytic continuation of f is an entire function satisfyingfor all z in the complex plane. Hence f is of at most order one and type one. In this note, we prove the following theorem.

scholarly journals Some representation formulae for entire functions of exponential type

1988 ◽  
Vol 37 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Entire Functions ◽  
Bernoulli Numbers ◽  
First Case ◽  
Functions Of Exponential Type ◽  
Explicit Formulae ◽  
Image Position ◽  
Fejér Mean ◽  
Derivatives Of

We obtain some explicit formulae for series of the typewhere f is an entire function of exponential type τ, bounded on the real exis (and satisfying in the first case). These series are expressed in terms of the derivatives of f and Bernoulli numbers. We examine the case where f is a trigonometric polynomial which lead us, in particular, to a new representation of the associated Fejér mean.

Inequalities for the derivatives of a bounded harmonic function

1948 ◽  
Vol 44 (2) ◽  
pp. 155-158 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Unit Circle ◽  
Harmonic Functions ◽  
Conformal Transformations ◽  
Alternative Method ◽  
Schwarz’S Lemma ◽  
The One ◽  
Image Position ◽  
Derivatives Of ◽  
Bounded Harmonic Function

If h(r, θ) is harmonic in the unit circle | r | < 1 and satisfies the condition | h | ≤ 1, then there is a function u(ø) which satisfies | u | ≤ 1 such thatand conversely. Hence, any properties of such harmonic functions should be deducible from equation (1). A number of such properties have been proved by Koebe (Math. Z. 6 (1920), 52–84, 69), using Schwarz's lemma and the geometry of simple conformal transformations. They can be deduced from (1) together with an elementary lemma on the rearrangement of a function (Lemma 1 below). As, however, students of this subject will regard Koebe's method as the one best adapted to establish his theorems, we shall illustrate the alternative method by considering two new problems, namely to find max ∂h/∂r, max ∂h/∂θ, where the maximum in each case is taken for all harmonic functions h which satisfy

scholarly journals A uniqueness theorem for harmonic functions on half-spaces

1989 ◽  
Vol 31 (2) ◽  
pp. 189-191 ◽  
Author(s):  
D. H. Armitage
Keyword(s):  
Euclidean Space ◽  
Uniqueness Theorem ◽  
Half Space ◽  
Harmonic Functions ◽  
Arbitrary Point ◽  
Euclidean Norm ◽  
Image Position

An arbitrary point of the Euclidean space Rn+1, where n > 1, is denoted by (X, y), where X ∈ Rn and y ∈ R, and we denote the Euclidean norm on Rn by ∥·∥. If h is harmonic on the half-space Ω = {(X, y): y > 0}, then we define extended real-valued functions m and M as follows:and

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

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