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Riku Klén

Publications and source records attributed to Riku Klén.

29 records · Page 2Linked to original sources
arXiv2013

Quasihyperbolic metric and Möbius transformations

An improved version of quasiinvariance property of the quasihyperbolic metric under Möbius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is given. Next, a quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is proved. Finally, several inequalities between the quasihyperbolic metric and other commonly used metrics such as the hyperbolic metric of the unit ball and the chordal metric are established.

math.CV↗
arXiv2013

Inclusion relations of hyperbolic type metric balls II

Inclusion relations of metric balls defined by the hyperbolic, the quasihyperbolic, the $j$-metric and the chordal metric will be studied. The hyperbolic metric, the quasihyperbolic metric and the $j$-metric are considered in the unit ball.

math.MG↗
arXiv2012

Subdomain geometry of hyperbolic type metrics

Given a domain $G \subsetneq \Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D \subset G$. In each case, distances in the subdomain are always larger than in the original domain. Our goal is to show that, in several cases, one can prove a stronger domain monotonicity statement. We also show that under special hypotheses we have inequalities in the opposite direction.

math.MG↗
arXiv2012

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional $p-$Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinović-Adamović inequality, Lazarević's inequality, Huygens-type inequalities, Wilker-type inequalities, and Cuza-Huygens-type inequalities, are generalized to the case of generalized functions.

math.CA↗
arXiv2012

Local convexity properties of Apollonian and Seittenranta's metric balls

We consider local convexity properties of balls in the Apollonian and Seittenranta's metrics. Balls in the Apollonian metric are considered in the twice punctured space and starlike domains. Balls in Seittenranta's metric are considered in the twice punctured space and in the punctured ball.

math.MG↗
arXiv2012

Dimension of the boundary in different metrics

We consider metrics on Euclidean domains $Ω\subset\R^n$ that are induced by continuous densities $ρ\colonΩ\rightarrow(0,\infty)$ and study the Hausdorff and packing dimensions of the boundary of $Ω$ with respect to these metrics.

math.CA↗
arXiv2012

Apollonian circles and hyperbolic geometry

The goal of this paper is to study two basic problems of hyperbolic geometry. The first problem is to compare the hyperbolic and Euclidean distances. The second problem is to find hyperbolic counterparts of some basic geometric constructions such as the construction of the middle point of a hyperbolic geodesic segment. Apollonian circles have a key role in this study.

math.MG↗
arXiv2011

Inclusion relations of hyperbolic type metric balls

We will consider inclusion of metric balls defined by the quasihyperbolic, the $j$-metric and the chordal metric. The quasihyperbolic metric and the $j$-metric are considered in general subdomains of $\mathbb{R}^n$ and in some particular domains like $\mathbb{R}^n \setminus {0}$ and the upper half-space.

math.MG↗
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

Close-to-convexity of quasihyperbolic and $j$-metric balls

We will consider close-to-convexity of the metric balls defined by the quasihyperbolic metric and the $j$-metric. We will show that the $j$-metric balls with small radii are close-to-convex in general subdomains of $\Rn$ and the quasihyperbolic balls with small radii are close-to-convex in the punctured space.

math.MG↗