Remarks on the Scale-Invariant Cassinian Metric
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Remarks on the Scale-Invariant Cassinian Metric Gendi Wang1 · Xiaoxue Xu1 · Matti Vuorinen2 Received: 16 March 2020 / Revised: 3 July 2020 / Accepted: 28 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We study the geometry of the scale-invariant Cassinian metric and prove sharp comparison inequalities between this metric and the hyperbolic metric in the case when the domain is either the unit ball or the upper half space. We also prove sharp distortion inequalities for the scale-invariant Cassinian metric under Möbius transformations. Keywords The scale-invariant Cassinian metric · The hyperbolic metric · Möbius transformation Mathematics Subject Classification 30F45 · 51M10
1 Introduction In the Euclidean space Rn , n ≥ 2 , the natural way to measure distance between two points x, y ∈ Rn is to use the length |x − y| of the segment joining the points. In geometric function theory [6], one studies functions defined in subdomains D ⊂ Rn , and measures distances between two points x, y ∈ D . In this case, the Euclidean distance is no longer an adequate method for measuring the distance, because one has to take into account also the position of the points relative to the boundary ∂ D . During the past few decades, many authors have suggested metrics for this purpose. In the case of the simplest domain, the unit ball Bn , we have the hyperbolic or Poincaré
Communicated by Saminathan Ponnusamy.
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Gendi Wang [email protected] Xiaoxue Xu [email protected] Matti Vuorinen [email protected]
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School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
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Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland
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G. Wang et al.
metric that is the most common metric in this case. Therefore, it is a natural idea to analyze the various equivalent definitions of the hyperbolic metric and to use these to generalize, if possible, the hyperbolic metric to the case of a given domain D ⊂ Rn . These generalizations capture usually some but not all features of the hyperbolic metric and are thus called hyperbolic type metrics [3,5,7,9,10,12,13,15,16,20]. Because the usefulness of a metric depends on how well its invariance properties match those of the function spaces studied, we now analyze hyperbolic type metrics from this point of view. The best we can expect is invariance in the same sense as the hyperbolic metrics are invariant, namely invariance under Möbius transformations n n of the Möbius space (R , q) , R = Rn ∪ {∞} , equipped with the chordal metric q . Another useful notion is invariance with respect to similarity transformations. A similarity transformation is a transformation of the form x → λU (x) + b where λ > 0 , b ∈ Rn , and U is an orthogonal map, i.e., a linear map with |U (x)| = |x| for all x ∈ Rn . The quasihyperbolic and the distance ratio metrics introduced by Gehring and Palka [7] have become widely used hyperbolic type metrics in geometric function theory in plane and space [6]. Both metrics are d
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