Proof of Poisson's Formula

1961 ◽  
Vol 57 (1) ◽  
pp. 186-186 ◽  
Author(s):  
Bengt Åkerberg
Keyword(s):  
Unit Circle ◽  
Harmonic Functions ◽  
Image Position

Let f(z)=u(x, y)+iv(x, y) be regular in a domain containing the unit circle C. If where r < 1, then is regular in |z| ≤ 1 and we have and taking real parts and imaginary parts we obtain Poisson's formulae for the harmonic functions u and v.

A theorem on positive harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 207-212 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Unit Circle ◽  
Harmonic Functions ◽  
Closed Interval ◽  
Positive Harmonic Function ◽  
Image Position ◽  
Function Conjugate ◽  
Positive Harmonic Functions

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can writewhere α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to writewhere h* is a harmonic function conjugate to h. Then the functionhas the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.

Mean values and classes of harmonic functions

1984 ◽  
Vol 96 (3) ◽  
pp. 501-505 ◽  
Author(s):  
Thomas Ramsey ◽  
Yitzhak Weit
Keyword(s):  
Unit Circle ◽  
Harmonic Functions ◽  
Borel Measure ◽  
Mean Value ◽  
Finite Complex ◽  
Mean Values ◽  
Complex Borel Measure ◽  
Generalized Mean ◽  
Image Position

Let μ be a finite complex Borel measure supported on the unit circle.In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form.and for all ξ ∈ , | ξ | = R for some fixed R > 0.

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

Functions regular in the unit circle

1956 ◽  
Vol 52 (1) ◽  
pp. 49-60 ◽  
Author(s):  
C. N. Linden ◽  
M. L. Cartwright
Keyword(s):  
Unit Circle ◽  
Positive Constant ◽  
Upper Bounds ◽  
Image Position

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

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.

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

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

scholarly journals Radial growth and exceptional sets for Cauchy–Stieltjes integrals

1994 ◽  
Vol 37 (1) ◽  
pp. 73-89 ◽  
Author(s):  
D. J. Hallenbeck ◽  
T. H. MacGregor
Keyword(s):  
Unit Circle ◽  
Radial Growth ◽  
Borel Measure ◽  
Local Conditions ◽  
Exceptional Sets ◽  
Image Position ◽  
Complex Valued

This paper considers the radial and nontangential growth of a function f given bywhere α>0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

The Isometries of Hp

1964 ◽  
Vol 16 ◽  
pp. 721-728 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Circle ◽  
Lebesgue Measure ◽  
Measurable Functions ◽  
Image Position ◽  
Complex Valued

Let a be the Lebesgue measure on the unit circle |z| = 1 withand let Lp be the space of complex-valued σ-measurable functions f such thatis finite. Hp is the closure in Lp of the algebra of analytic polynomials

scholarly journals Local uniqueness in boundary problems

1970 ◽  
Vol 17 (1) ◽  
pp. 23-36
Author(s):  
M. H. Martin
Keyword(s):  
Incompressible Fluid ◽  
Unit Circle ◽  
Finite Amplitude ◽  
Local Uniqueness ◽  
Boundary Problems ◽  
Infinite Depth ◽  
Image Position

The study of periodic, irrotational waves of finite amplitude in an incompressible fluid of infinite depth was reduced by Levi-Civita (1) to the determination of a functionregular analytic in the interior of the unit circle ρ = 1 and which satisfies the condition

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