Q‐Type Spaces of Harmonic Mappings

The aim of this paper is twofold. First, we investigate the properties of the composition of harmonic mappings with harmonic mappings, and the composition of biharmonic mappings with harmonic mappings. Second, we consider the Goodman-Saff conjecture for biharmonic mappings in the unit disk. In fact, we show that the answer to the Goodman-Saff conjecture is positive for a special class of univalently biharmonic mappings which contains the set of all harmonic univalent mappings.

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ℝ n are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejér-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.

Isoperimetric type inequalities for harmonic functions

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \\overline {g} $ where h and g are analytic functions. If $ f \\in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \\in K_H^0 $ . For $ f \\in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\ ilde {*}\\varphi = h*\\varphi + \\overline {g*\\varphi } $ is found. With } a strip mapping, $ f\ ilde {*}\\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \\in K_H^0 $ which take the unit disk to regular n -gons.

The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ⩽ ρ < 1, the author constructs a closed convex domain E r,ρ such that $$F\left( {\bar \Delta \left( {z,r} \right)} \right) \subset e^{i\alpha } E_{r,\rho } = \left\{ {e^{i\alpha } z:z \in E_{r,\rho } } \right\}$$ holds for every z ∈ D, w = ρe iα and harmonic mapping F with F(D) ⊂ D and F(z) = w, where Δ(z, r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρe iα and w′ ∈ e iα E r,ρ , there exists a harmonic mapping F such that F(D) ⊂ D, F(z) = w and F(z′) = w′ for some z′ ∈ ∂Δ(z, r). (II) The author establishes a Finsler metric H z (u) on the unit disk D such that $$\mathcal{H}_{F\left( z \right)} \left( {e^{i\theta } F_z \left( z \right) + e^{ - i\theta } F_{\bar z} \left( z \right)} \right) \leqslant \frac{1} {{1 - \left| z \right|^2 }}$$ holds for any z ∈ D, 0 ⩽ θ ⩽ 2π and harmonic mapping F with F(D) ⊂ D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg\(\left( {e^{i\theta } Fz\left( z \right) + e^{ - i\theta } F_{\bar z} \left( z \right)} \right)\).

In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic -like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas’yanyuk [10] and Brickman [3] for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mappings with the help of an improved distortion theorem.

In this paper the of Rad o-Kneser-Choquet is extended in two dierent ways to multiply connected domains. One is a direct continuation of Kneser’s idea and has nothing to do with convexity; while the other asserts that a nitely connected domain can be mapped harmonically with prescribed outer boundary correspondence onto a given convex domain with suitable punctures. It is also shown that a domain containing innity admits a unique harmonic mapping, with standard normalization at innity, onto a punctured plane. For domains of connectivity n the dilatation of the canonical mapping covers the unit disk exactly 2n times. Furthermore, no other normalized harmonic mapping has the same dilatation. In 1926, T. Rad o[ 22] posed the problem to show that for any homeomorphism of the unit circle onto the boundary @ of a bounded convex domain ; the harmonic extension f maps the unit disk D univalently onto : In response, H. Kneser [14] supplied an elegant proof. Some 20 years later G. Choquet [4], apparently unaware that the was known, rediscovered it and gave another proof. Fortunately, the two proofs are dierent and even for simply connected domains they have dierent generalizations. The dichotomy between the two approaches of Kneser and Choquet comes into sharper focus as the is generalized to multiply connected domains. In presenting these generalizations, it will be expedient to distinguish between \Kneser’s theorem and \Choquet’s Kneser’s proof has little to do with convexity, while Choquet’s proof uses convexity in a more essential way. Indeed, Kneser’s proof applies (as he indicates in [14]) when is not convex, under the additional hypothesis that f(D) : We shall see that the main idea of his proof carries over to multiply connected domains. On other hand, by methods more akin to Choquet’s proof we will show that a nitely connected domain D can be mapped harmonically, with prescribed boundary values, onto a given convex domain with punctures at suitable points. Another result is that D can be mapped harmonically onto a punctured plane, and such a mapping is unique up to a normalization. Our proofs adapt an idea of Clunie and Sheil-Small [5], which gives yet another proof of the Rad o-Kneser-Choquet theorem.

In this paper we study the distribution of values of harmonic functions in non-Euclidean disks. We introduce the notion of a P′-sequence, which enables us to characterize the class of normal harmonic functions defined in the unit disk. Moreover, we obtain sufficient conditions for the existence of such sequences and give examples which show that these conditions are essential in the stated theorems.

Complex valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form f = h + g ,w hereh and g are analytic in U. In this paper, we introduce a class HP(α), (α 0) of functions which are harmonic in U. We give sufficient coefficient conditions for normalized harmonic functions in HP(α). These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.

The theory of harmonic spaces was mainly established with the aim to generalize and unify results and methods of classical potential theory for application to an extended class of elliptic and parabolic differential equations of second order. Originally, the theory started with a sheaf of vector spaces of real continuous functions on a locally compact space, playing the role of the sheaf of solutions of a partial differential equation. A convergence property, the boundary minimum principle, the local solvability of the Dirichlet problem are supposed to hold. The most important type of a harmonic space is a Bauer space which is introduced in section 1. In our terminology a Bauer space is locally a harmonic space having a base of regular sets. Semi-elliptic differential operators are treated in section 2. In section 3 we present J. M. BONY’s result that a Bauer space whose harmonic functions are smooth is generated by such a differential operator. Section 4 prepares the material (Sobolev spaces, weak solutions, etc.) needed in section 5 to show that elliptic-parabolic differential operators generate Bauer spaces. Besides the deep result of L. HORMANDER on the hypoellipticity of such operators the theory is completely selfcontained. For sake of simplicity we mostly assume that the constant function 1 is harmonic.KeywordsWeak SolutionDifferential OperatorHarmonic FunctionOpen SubsetDirichlet FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Radius of convexity of partial sums of functions in the close-to-convex family

Circle companions of Hardy spaces of the unit disk

A set E c Q is holomorphically dominating for Q if sUP,cE If(z)I = SUpzcn If(z)l for all holomorphic functions f on Q. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point * (of Q) from Q x E. Moreover, it is equivalent to a large number of other statements (old and new) of holomorphic, harmonic and topological nature, including that a certain weighted Bergman space with p = oo is a Banach space. We extend this to the cases of harmonic functions in R' and holomorphic functions in cV. We also present some results on when weighted Bergman spaces are (quasi)-Banach spaces, the case p = oo being characterised by the result mentioned above.

The generalized growth of entire transcendental functions in terms of polynomial approximation errors in some Banach spaces has been studied by various authors. The main purpose of this paper is to study the harmonic polynomial approximation of entire harmonic functions in space ℝn, n ≥ 3, in certain Banach spaces. Moreover, the generalized type of harmonic functions of slow growth has been characterized in terms of best harmonic polynomial approximation errors. Our results add new aspects for the case of order zero.

We define Hardy spaces of functions taking values on a Banach space X over nonsmooth domains. The types of functions we consider are harmonic functions on a starlike Lipschitz domain and solutions to the heat equation on a time-varying domain. Our purpose is twofold: (a) to characterize the Radon–Nikodym property of the Banach space X in terms of the existence of nontangential limits of X-valued functions u in the corresponding Hardy space with index p≥1, (b) to identify the function of the boundary values of u in the Hardy space with index p>1 with an element in the space VXp of measures of p-bounded variation in the absence of the Radon–Nikodym property of X. This extends similar results already known on the unit disk of C and the semispace Rn×(0,∞).