A theorem on 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.

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A note on positive harmonic functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100024257 ◽

1948 ◽

Vol 44 (2) ◽

pp. 289-291 ◽

Cited By ~ 1

Author(s):

S. Verblunsky

Keyword(s):

Harmonic Function ◽

Harmonic Functions ◽

Duke Math ◽

Negative Number ◽

Image Position ◽

American Math ◽

Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

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On Positive Harmonic Functions in a Half-Plane

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100013499 ◽

1935 ◽

Vol 31 (4) ◽

pp. 482-507 ◽

Cited By ~ 1

Author(s):

S. Verblunsky

Keyword(s):

Harmonic Function ◽

Half Plane ◽

Harmonic Functions ◽

Negative Number ◽

Cartesian Coordinates ◽

Image Position ◽

Positive Harmonic Functions

1. Let ξ, η denote the rectangular Cartesian coordinates of a point in a plane. Let J (ξ, η) denote a harmonic function which is positive in the half-plane η > 0. In this paper, we first show (Theorem I) that every such function J determines a non-negative number d, and a bounded non-diminishing function G(x), such that

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Inequalities for the derivatives of a bounded harmonic function

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100024129 ◽

1948 ◽

Vol 44 (2) ◽

pp. 155-158 ◽

Cited By ~ 1

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

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V.—On Whittaker's Solution of Laplace's Equation

Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences ◽

10.1017/s0080454100006397 ◽

1944 ◽

Vol 62 (1) ◽

pp. 31-36

Author(s):

E. T. Copson

Keyword(s):

Harmonic Function ◽

General Solution ◽

Arbitrary Function ◽

Harmonic Functions ◽

Singular Points ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Image Position

In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the formwhere f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.

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Minimum growth of harmonic functions and thinness of a set

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100061363 ◽

1984 ◽

Vol 95 (1) ◽

pp. 123-133 ◽

Cited By ~ 1

Author(s):

Jang-Mei G. Wu

Keyword(s):

Harmonic Function ◽

Harmonic Functions ◽

Integral Condition ◽

Finite Point ◽

Image Position ◽

Increasing Function

In [3], Barth, Brannan and Hayman proved that if u(z) is any non-constant harmonic function in ℝ2, ø(r) is a positive increasing function of r for r ≥ 1 andthen there exists a path going from a finite point to ∞, such that u(z) > ø(|z|) on Γ. Moreover, they showed by example that the integral condition above cannot be relaxed.

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Mean values and classes of harmonic functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100062435 ◽

1984 ◽

Vol 96 (3) ◽

pp. 501-505 ◽

Cited By ~ 4

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.

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Domination of the supremum of a bounded harmonic function by its supremum over a countable subset

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500026869 ◽

1987 ◽

Vol 30 (3) ◽

pp. 471-477 ◽

Cited By ~ 9

Author(s):

F. F. Bonsall

Keyword(s):

Harmonic Function ◽

Unit Disc ◽

Harmonic Functions ◽

Open Unit ◽

Countable Subset ◽

Open Unit Disc ◽

Image Position ◽

Bounded Harmonic Functions ◽

Bounded Harmonic Function

For what sequences {an} of points of the open unit disc D does there exist a constant k such thatfor all bounded harmonic functions f on D?

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Asymptotic tracts of harmonic functions II

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500006180 ◽

1995 ◽

Vol 38 (1) ◽

pp. 35-52 ◽

Cited By ~ 1

Author(s):

K. F. Barth ◽

D. A. Brannan

Keyword(s):

Harmonic Function ◽

Real Number ◽

Real Function ◽

Connected Domain ◽

Harmonic Functions ◽

Simply Connected Domain ◽

Image Position ◽

Simply Connected ◽

Prime End

An asymptotic tract of a real function u harmonic and non-constant in ℂ is a component of the set {z:u(z)≠c}, for some real number c; a quasi-tractT(≠ℂ) is an unbounded simply-connected domain in ℂ such that there exists a function u that is positive, unbounded and harmonic in T such that, for each point ζ∈∂T∩ℂ,and a ℱ-tract is an unbounded simply-connected domain T in ℂ whose every prime end that contains ∞ in its impression is of the first kind.The authors study the growth of a harmonic function in one of its asymptotic tracts, and the question of whether a quasi-tract is an asymptotic tract. The branching of either type of tract is also taken into consideration.

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A harmonic quadrature formula characterizing open strips

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075836 ◽

1993 ◽

Vol 113 (1) ◽

pp. 147-151 ◽

Cited By ~ 2

Author(s):

D. H. Armitage ◽

C. S. Nelson

Keyword(s):

Harmonic Function ◽

Lebesgue Measure ◽

Quadrature Formula ◽

Open Subset ◽

Harmonic Functions ◽

Mean Value ◽

Open Ball ◽

Mean Value Property ◽

Image Position ◽

Dimensional Lebesgue Measure

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, thenA converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such thatfor every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.

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Proof of Poisson's Formula

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100035015 ◽

1961 ◽

Vol 57 (1) ◽

pp. 186-186 ◽

Cited By ~ 1

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.

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