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Criteria of univalence and fully \alpha--accessibility for p--harmonic and p--analytic functions
Abstract
In this article, we present univalence criteria for polyharmonic and polyanalytic functions. Our approach yields a new criterion for a polyharmonic functions to be fully
-accessible. Several examples are presented to illustrate the use of these criteria.
... .} (cf. [2,5,9,27] ...
... ). Recently, several new properties of polyharmonic mappings are discussed in[2]. The following result concerns the solutions to the equation (1.3).Lemma C [25, Theorem 2.2] Let α ∈ R and f ∈ C 2 (D). ...
The main aim of this paper is to study the Lipschitz continuity of certain (K,K′)-quasiconformal mappings with respect to the distance ratio metric, and the Lipschitz continuity of the solution of a quasilinear differential equation with respect to the distance ratio metric.
... .} (cf. [1,2,5,6,11,13,15,27]). In particular, if α = 0, then u is harmonic (cf. ...
We call the solution of a kind of second order homogeneous partial differential equation as real kernel α-harmonic mappings. In this paper, the representation theorem, the Lipschitz continuity, the univalency and the related problems of the real kernel α-harmonic mappings are explored.
... Using the work of Abdulhadi and AbuMuhanna [2], several authors (eg. Chen et al., Liu et al.) investigated Landau-Bloch type theorems for polyharmonic mapping with different normalizations (see [5,9,13,20]). We first give a version of Landau's theorem for poly-analytic functions. ...
A continuous complex-valued function F in a domain D⊆C is poly-analytic of order α if it satisfies ∂αF/∂z¯α=0. One can show that F has the form F(z)=∑k=0α-1z¯kAk(z), where each Ak is an analytic function. In this paper, we prove the existence of a Landau constant for poly-analytic functions and the special bi-analytic case. We also establish Bohr’s inequality for poly-analytic and bi-analytic functions. In addition, we give an estimate for the arc-length over the class of poly-analytic mappings and consider the problem of minimizing moments of order p.
... Investigation of biharmonic mappings in the context of the geometric function theory is started only recently. Indeed several important properties of biharmonic mappings are obtained in [1][2][3][4]6,14] and these mappings were also generalized by some others, see for example [9,27] and the references therein. Biharmonic mappings arise naturally in fluid dynamics and elasticity problems, and have important applications in engineering and biology (see [21,25]). ...
In this paper, we shall discuss the family of biharmonic mappings for which maximum principle holds. As a consequence of our study, we present Schwarz Lemma for the family of biharmonic mappings. Also we discuss the univalency of certain class of biharmonic mappings.
As a generalization of conformal mappings and planar harmonic mappings, biharmonic mappings have attracted more and more attention. This paper investigates the properties of biharmonic mappings. First, in terms of coefficients, we give sufficient conditions for quasiconformal extensions of biharmonic mappings in the unit disk. Second, we consider a class of strongly starlike biharmonic mappings called hereditarily strongly starlike biharmonic mappings. We obtain quasiconformal extensions of hereditarily strongly starlike biharmonic mappings under a condition on the dilatations. Based on the coefficients and convolution, respectively, we then give two sufficient conditions for hereditarily strongly starlike biharmonic mappings. Finally, we study quasiconformal extensions of biharmonic mappings in the exterior unit disk.
In this paper, we study Schwarz lemma for solutions of the α-harmonic equation. Applying the method given by Burgeth, we build a new form of Schwarz lemma for solutions of the α-harmonic equation, which is suitable for α>-1. Moreover, we generalize the Schwarz–Pick inequality for solutions of the α-harmonic equation from the case α≥0 to α>-1.
We call the solution of a kind of weighted Laplace equation complex-valued kernel -harmonic mappings. In this paper, some geometric properties of the complex-valued kernel -harmonic mappings, such as area, fully starlikeness and fully convexity of order , , are explored. Furthermore, for a class of given boundary function, the Radó–Kneser–Choquet-type theorem is obtained.
The paper considers the domains with cone condition
in C. We say that domain G satisfies the (weak) cone condition, if
p+V (e(p), H) ⊂ G for all p ∈ G, where V (e(p), H) denotes rightangled
circular cone with vertex at the origin, a fixed solution ε
and a height H, 0 < H ≤ ∞, and depending on the p vector e(p)
axis direction.
Domains satisfying cone condition play an important role in
various branches of mathematic (e. g. [1], [2], [3] (p. 1076), [4]).
In the paper of P. Liczberski and V. V. Starkov, α–accessible
domains were considered, α ∈ [0, 1), — the domains, accessible
at every boundary point by the cone with symmetry axis on
{pt : t > 1}.
Unlike the paper of P. Liczberski and V. V. Starkov, here we
consider domains, accessible outside by the cone, which symmetry
axis inclined on fixed angle ϕ to the {pt : t > 1}, 0 < ∥ϕ∥ <
< π/2.
In this paper we give criteria for this class of domains when
the boundaries of domains are smooth, and also give a sufficient
condition when boundary is arbitrary.
This article is the full variant of [5], published without proofs.
In [1], [2] (α, β)-accessible domains in C were defined and investigated, and the criterion of (α, β)-accessibility in a smooth case was obtained. (α, β)-accessible domains are starlike, γ-accessible (γ = min{α, β}), and satisfy the so-called “cone condition” (i. e. the domains are conically accessible from the interior), which is important for applications, such as the theory of integral representations of functions, imbedding theorems, the questions of the boundary behavior of functions, the solvability of Dirichlet problem. In this paper the author obtains the necessary and some sufficient conditions of (α, β)-accessibility of domain in nonsmooth case.
The criteria of the univalence of a harmonic mapping is obtained in this paper. Particularly, it permits to formulate the hypothesis of the harmonic function classes equality S 0 H = S 0 H(S) (hypothesis of Ponnusamy and Sairam), in analytic form. The method of construction of the univalent harmonic polynomials with desired properties, according to a given harmonic function, is obtained by means of the univalence criteria.
A 2p-times continuously differentiable complex-valued function F = u + iv in a domain Ω ⊆ C is p-harmonic if F satisfies the p-harmonic equation ∆ p F = 0. We say that F is log-p-harmonic if log F is p-harmonic. In this paper, we investigate several ba-sic properties of p-harmonic and log-p-harmonic mappings. In particular, we discuss the problem of when the composite mappings of p-harmonic mappings with a fixed analytic function are q-harmonic, where q ∈ {1, . . . , p}. Also, we obtain necessary and sufficient conditions for a function to be p-harmonic (resp. log-p-harmonic). We study the local univalence of p-harmonic and log-p-harmonic mappings, and in particular, we obtain two sufficient conditions for a function to be a locally univalent p-harmonic or a locally univalent log-p-harmonic. The starlikeness of log-p-harmonic mappings is considered.
A geometric interpretation of the Schwarzian of a harmonic map-ping is given in terms of geodesic curvature on the associated minimal surface, generalizing a classical formula for analytic functions. A formula for curvature of image arcs under harmonic mappings is applied to derive a known result on concavity of the boundary. It is also used to characterize fully convex mappings, which are related to fully starlike mappings through a harmonic analogue of Alexander's theorem.
A 2p-times continuously differentiable complex-valued function f=u+iv in
a simply connected domain is \textit{p-harmonic} if
f satisfies the p-harmonic equation In this paper, we
investigate the properties of p-harmonic mappings in the unit disk .
First, we discuss the convexity, the starlikeness and the region of variability
of some classes of p-harmonic mappings. Then we prove the existence of Landau
constant for the class of functions of the form Df=zf_{z}-\barzf_{\barz},
where f is p-harmonic in . Also, we discuss the region of
variability for certain p-harmonic mappings. At the end, as a consequence of
the earlier results of the authors, we present explicit upper estimates for
Bloch norm for bi- and tri-harmonic mappings.
Polyanalytic functions emerged in the mathematical theory of elasticity: eighty years after the discovery of its basic equations, Kolossoff found that functions of the form φ(z) +
Ψ(z), where φ and Ψ are analytic functions, can be an efficient tool for solving problems of the planar theory of elasticity. Functions of this form were later called bianalytic. Useful applications of this idea in mechanics are widely known from the remarkable investigations by Kolossoff, Muskhelishvili and their followers. The class of polyanalytic functions is an extension of the class of bianalytic functions. This survey is devoted to the former class.
The class A
ρα of domains D ⊂ ℝn which are α-accessible with respect to the origin, α ∈ [0, 1], and have the property , where ρ ∈ (0, +∞) is a fixed number, is considered. Such domains satisfy the so-called cone condition, i.e., are conically accessible from the interior. The maximal set of points a such that all domains D ∈ A
ρα are β-accessible with respect to a, 0 ≤ β ≤ α, is found, which solves a problem posed by Professor S.I. Dudov. This set is proved to be the open ball of radius ρ centered at 0 if α = 1 and β = 0 and the closed ball of radius centered at 0 otherwise. An answer to a question of Professor S. R. Nasyrov is also given. For domains D ⊈ ℝn
α-accessible with respect to the origin, α ∈ (0, 1], a sharp upper bound for is found.
In this paper, we introduce a scale of differential operators which is shown to correspond canonically to a certain scale of solution kernels generalizing the classical Poisson kernel for the unit disc. The scale of kernels studied is very natural and appears in many places in mathematical analysis, such as in the theory of integral representations of biharmonic functions in the unit disc.
In this paper some characterizations and properties of domains in Re with boundary accessible externally by some circular cones are given. It is shown that the boundary of such domains is also conically accessible from the interior. It is also characterized alpha-accessible domains of C-N, which are biholomorphically equivalent to the open Euclidean ball. The paper is also a continuation of a concept of V.V. Starkov (2011) [13] and refers to papers of D.A. Brannan and W.E. Kirwan (1969) [3], W. Ma and D. Minda (1991) [10], J. Stankiewicz (1966) [11], and T. Sugawa (2005) [15] connected with a planar case.
We prove Schwarz-Pick type estimates and coefficient estimates for a class of
elliptic partial differential operators introduced by Olofsson. Then we apply
these results to obtain a Landau type theorem.
In the paper we present domains in C with special cone condition.
A 2p-times continuously differentiable complex-valued function f = u+iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation Δpf = 0, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂn is a domain, then a function f: Ω → ℂm is said to be p-harmonic in Ω if each component function fi (i ∈ {1,...,m}) of f = (f1,...; fm) is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch's theorem for a class of p-harmonic mappings f from the unit ball Bn into ℂn with the form where each Gp-k1+1,..., p-kn+1 is harmonic in Bn for ki ∈ {1,...,p} and i ∈ {1,...,n}.
1. Preliminaries 2. Local properties of harmonic mappings 3. Harmonic mappings onto convex regions 4. Harmonic self-mappings of the disk 5. Harmonic univalent functions 6. Extremal problems 7. Mapping problems 8. Additional topics 9. Minimal surfaces 10. Curvature of minimal surfaces Appendix References.
In this paper we first determine conditions on the analytic polynomials pp and qq such that the family of univalent sense-preserving biharmonic polynomials of the form P=|z|2(p+q¯) is dense in the family of univalent sense-preserving biharmonic mappings FF defined on the unit disk DD of the form F=|z|2GF=|z|2G. Next, we consider the univalency of harmonic function GG whenever F=|z|2GF=|z|2G is univalent in DD. Finally, we give a partial answer to the problem raised by Muhanna about the radius of univalence of a family of sense-preserving biharmonic mappings with the form F=|z|2G+HF=|z|2G+H, where GG is univalent in DD.
In this paper, our main aim is to introduce the concept of planar p-harmonic mappings and investigate the properties of these mappings. First, we discuss the p-harmonic Bloch mappings. Two estimates on the Bloch constant are obtained, which are generalizations of the main results in Colonna (1989) [9]. As a consequence of these investigations, we establish a Bloch and Landau's theorem for p-harmonic mappings.
To address the uniqueness issues associated with the Dirichlet problem for
the N-harmonic equation on the unit disk \D in the plane, we investigate
the integrability of N-harmonic functions with respect to the standard
weights . The question at hand is the following. If u
solves in \D, where stands for the Laplacian, and
[\int_\D|u(z)|^p (1-|z|^2)^{\alpha}\diff A(z)<+\infty,] must then
? Here, N is a positive integer, is real, and
; \diff A is the usual area element. The answer will, generally
speaking, depend on the triple . The most interesting case is
. For a given N, we find an explicit critical curve
-- a piecewise affine function -- such that for
there exist non-trivial functions u with
of the given integrability, while for , only
is possible. We also investigate the obstruction to uniqueness for the
Dirichlet problem, that is, we study the structure of the functions in
\mathrm{PH}^p_{N,\alpha}(\D) when this space is nontrivial. We find a
fascinating structural decomposition of the polyharmonic functions -- the
cellular (Almansi) expansion -- which decomposes the polyharmonic weighted
in a canonical fashion. Corresponding to the cellular expansion is a
tiling of part of the plane into cells. A particularly interesting
collection of cells form the entangled region.
We study the dependence between various sets of limiting boundary values of an arbitrary function
f of z. In particular we show that the Plesner theorem on angular boundary values of analytic
functions generalizes in a natural way to arbitrary functions.
The Sobolev imbedding theorem and certain interpolation inequalities for Sobolev spaces are established for a wider class of domains than has been covered by earlier proofs. This class is defined by a weakened, measure theoretic version of the cone condition. The proofs are elementary.
In this note, we will show that a simply connected bounded domain D is strongly starlike of order α < 1 with respect to the origin if and only if so is D∨, where D∨ is the analytic inversion of the exterior of D, namely, . This fact neatly explains the relationship between some known properties of strongly starlike domains and provides several new characterizations for those domains.
University lecture series
- L V Ahlfors
L. V. Ahlfors, Lectures on quasiconformal mappings, American Mathematical Society University lecture series, 38(2006), 164 p.