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Antti Rasila

Publications and source records attributed to Antti Rasila.

At least 73 records · Page 4Linked to original sources
arXiv2013

On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces

In this paper, we establish a Bloch-type growth theorem for generalized Bloch-type spaces and discuss relationships between Dirichlet-type spaces and Hardy-type spaces on certain classes of complex-valued functions. Then we present some applications to non-homogeneous Yukawa PDEs. We also consider some properties of the Lipschitz-type spaces on certain classes of complex-valued functions. Finally, we will study a class of composition operators on these spaces.

math.CV↗
arXiv2013

Coefficient estimates and the Fekete-Szeg\H o problem for certain classes of polyharmonic mappings

We give coefficient estimates for a class of close-to-convex harmonic mappings, and discuss the Fekete-Szegő problem of it. We also introduce two classes of polyharmonic mappings $\mathcal{HS}_{p}$ and $\mathcal{HC}_{p}$, consider the starlikeness and convexity of them, and obtain coefficient estimates on them. Finally, we give a necessary condition for a mapping $F$ to be in the class $\mathcal{HC}_{p}$.

math.CV↗
arXiv2013

Lengths, areas and Lipschitz-type spaces of planar harmonic mappings

In this paper, we establish a three circles type theorem, involving the harmonic area function, for harmonic mappings. Also, we give bounds for length and area distortion for harmonic quasiconformal mappings. Finally, we will study certain Lipschitz-type spaces on harmonic mappings.

math.CV↗
arXiv2013

Landau's theorem for polyharmonic mappings

In this paper, we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk $\mathbb{D}$. Then, we obtain two versions of Landau's theorem for polyharmonics mapping $F$, and for the mappings of the type $L(F)$, where $L$ is the differential operator of Abdulhadi, Abu Muhanna and Khuri. Examples and numerical estimates are given.

math.CV↗
arXiv2013

Starlikeness and convexity of polyharmonic mappings

In this paper, we first find an estimate for the range of polyharmonic mappings in the class $HC_{p}^{0}$. Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order $α$, and convex of order $β$, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions.

math.CV↗
arXiv2013

On polyharmonic univalent mappings

In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by $HS_{p}(λ)$, and its subclass $HS_{p}^{0}(λ)$, where $λ\in [0,1]$ is a constant. These classes are natural generalizations of a class of mappings studied by Goodman in 1950's. We generalize the main results of Avci and Złotkiewicz from 1990's to the classes $HS_{p}(λ)$ and $HS_{p}^{0}(λ)$, showing that the mappings in $HS_{p}(λ)$ are univalent and sense preserving. We also prove that the mappings in $HS_{p}^{0}(λ)$ are starlike with respect to the origin, and characterize the extremal points of the above classes.

math.CV↗
arXiv2013

On Quasihyperbolic Geodesics in Banach Spaces

We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex domain $Ω$ in a Banach space such that there is no geodesic between any given pair of points $x, y \in Ω\,.$ In addition, we prove that if $\mathrm{X}$ is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of $\mathrm{X}$, is smooth.

math.CV↗
arXiv2012

Mapping problems for quasiregular mappings

We study images of the unit ball under certain special classes of quasiregular mappings. For homeomorphic, i.e., quasiconformal mappings problems of this type have been studied extensively in the literature. In this paper we also consider non-homeomorphic quasiregular mappings. In particular, we study (topologically) closed quasiregular mappings originating from the work of J. Väisälä and M. Vuorinen in 1970's. Such mappings need not be one-to-one but they still share many properties of quasiconformal mappings. The global behavior of closed quasiregular mappings is similar to the local behavior of quasiregular mappings restricted to a so-called normal domain.

math.CV↗
arXiv2012

Computation of exterior moduli of quadrilaterals

We study the problem of computing the exterior modulus of a bounded quadrilateral. We reduce this problem to the numerical solution of the Dirichlet-Neumann problem for the Laplace equation. Several experimental results, with error estimates, are reported. Our main method makes use of an $hp$-FEM algorithm, which enables computations in the case of complicated geometry. For simple geometries, good agreement with computational results based on the SC Toolbox, is observed. We also use the reciprocal error estimation method introduced in our earlier paper to validate our numerical results. In particular, exponential convergence, in accordance with the theory of Babu\vska and Guo, is demonstrated.

math.NA↗
arXiv2012

Radial growth, Lipschitz and Dirichlet spaces on solutions to the Yukawa equation

In this paper, we investigate some properties to solutions $f$ to the Yukawa PDE: $Δf=λf$ in the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$, where $λ$ is a nonnegative constant. First, we prove that the answer to an open problem of Girela and Peláez, concerning such solutions, is positive. Then we study relationships on such solutions between the bounded mean oscillation and Lipschitz-type spaces. At last, we discuss Dirichlet-type energy integrals on such solutions in the unit ball of $\mathbb{C}^{n}$ and give an application.

math.AP↗
arXiv2012

Conjugate Function Method for Numerical Conformal Mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.

math.NA↗
arXiv2012

Harmonic Shears of Slit and Polygonal Mappings

In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations ω. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using Mathematica.

math.CV↗
arXiv2011

Quasihyperbolic Geometry in Euclidean and Banach Spaces

We consider the quasihyperbolic metric, and its generalizations in both the $n$-dimensional Euclidean space $R^n$, and in Banach spaces. Historical background, applications, and our recent work on convexity properties of these metrics are discussed.

math.CV↗
arXiv2010

On moduli of rings and quadrilaterals: algorithms and experiments

Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds.

math.NA↗