The Schwarz-Pick lemma for planar 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.

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic 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.

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.

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.

Suppose $$p\ge 1$$ , $$w=P[F]$$ is a harmonic mapping of the unit disk $${\mathbb D}$$ satisfying F is absolutely continuous and $${\dot{F}}\in L^p(0, 2\pi )$$ , where $${\dot{F}}(e^{it})=\frac{\mathrm {d}}{\mathrm {d}t}F(e^{it})$$ . In this paper, we obtain Bergman norm estimates of the partial derivatives for w, i.e., $$\Vert w_z\Vert _{L^p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{L^p}$$ , where $$1\le p<2$$ . Furthermore, if w is a harmonic quasiregular mapping of $${\mathbb {D}}$$ , then we show that $$w_z$$ and $$\overline{w_{{\bar{z}}}}$$ are in the Hardy space $$H^p$$ , where $$1\le p\le \infty $$ . The corresponding Hardy norm estimates, $$\Vert w_z\Vert _{p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{p}$$ , are also obtained.

We study the structure of inductive limits of weighted spaces of harmonic and holomorphic functions defined on the open unit disk of ℂ, and of the associated weighted locally convex spaces. Using a result of Lusky we prove, for certain radial weights on the open unit disk D of ℂ, that the spaces of harmonic and holomorphic functions are isomorphic to complemented subspaces of the corresponding Köthe sequence spaces. We also study the spaces of harmonic functions for certain non-radial weights on D. We show, under a natural sufficient condition for the weights, that the spaces of harmonic functions on D are isomorphic to corresponding spaces of continuous or bounded functions on ∂D.

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.

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

We give an overview of several conditions characterizing chord-arc and Ahlfors regular curves, Bishop-Jones domains and Bishop-Jones quasicircles in terms of Riemann mapping from the unit disk, welding map and, in case of quasicircles, quasiconformal Beltrami differential of an extension. We also consider generalizations of these results to harmonic quasiconformal mappings of unit disk to domain of corresponding type and pose several open problems.

The class $S_{H}(p)$, $0 \\leq p \\lt 1$, of complex valued, meromorphic harmonic univalent sense-preserving functions in the unit disk $U \\backslash \\{p\\}$ is studied. The functions belong to $S_{H}(p)$ have the expansion $f(z) = \\frac{\\alpha}{z-p} + \\sum_{n=0}^{\\infty} c_{n} z^{n} + \\overline{\\sum_{n=1}^{\\infty} d_{n} z^{n}} + A\\log |z-p|$ and $\\lim_{z \\rightarrow p} f(z) = \\infty$. Some coefficient estimates, distortion and area theorems are obtained. Sufficient coefficient conditions for a class of meromorphic harmonic univalent sense-preserving functions that are starlike and convex are given.

By applying Ruscheweyh derivative on the class ASH(λ, α, k, γ )o f harmonic univalent functions in the unit disk U , we obtain several interesting properties such as sharp coefficient relations, distortion bounds, extreme points, Hadamard product, and other results.

In this paper, using Differential operator a certain subclass of harmonic univalent functions in the unit disc U = {z ∈ C : |z| &lt; 1} is investigated. Some properties such as coefficient bounds, convex combination, extreme points and convolution conditions of this class are obtained. The newly introduced subclass of harmonic univalent functions is more general than the subclasses earlier found in the literature.

In this paper, we attempt to investigate a new harmonic mapping class denoted by the Banach space in the unit disk. Several characteristics of the class are examined. We additionally describe some findings for the little harmonic space, which really is a closed subspace of . We also talk about the boundedness and compactness of the composition operator in the space.