Antti Rasila

Convexity properties of quasihyperbolic balls on Banach spaces

We study the convexity and starlikeness of metric balls on Banach spaces when the metric is the quasihyperbolic metric or the distance ratio metric. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered.

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 Kuhnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North-Holland, Amsterdam, 2005]. 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

ON QUASIHYPERBOLIC GEODESICS IN BANACH SPACES

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Conjugate function method for numerical conformal mappings

-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

Coefficient estimates, Landau's theorem and Lipschitz-type spaces on planar harmonic mappings

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Harmonic shears of slit and polygonal mappings

-FEM algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds.

Yukawa Potential, Panharmonic Measure and Brownian Motion

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 quasihy- perbolic geodesics are unique. We also give an example of a convex domain in a Banach space such that there is no geodesic between any given pair of points x,y ∈ . In addition, we prove that if 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 X, is smooth.

Coefficient estimates and radii problems for certain classes of polyharmonic 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.

Harmonic close-to-convex functions and minimal surfaces

In this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss the coecient estimates and Landaus Theorem for some classes of locally univalent harmonic mappings, and then we study some Lipschitz-type spaces for locally univalent and multivalent harmonic mappings.

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

Abstract 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 .