Abstract. The Cassinian metric and its inner metric have been studied for subdomains of the n-dimensional Euclidean space R n (n ≥ 2) by the first named author. In this paper we obtain various inequalities between the Cassinian metric and other related metrics in some specific subdomains of R n . Also, a sharp distortion property of the Cassinian metric under Möbius transformations of the unit ball is obtained.2010 Mathematics Subject Classification. 30C35, 30C20, 30F45, 51M10. Key words and phrases. Möbius transformation, the hyperbolic metric, the Cassinian metric, the distance ratio metric, the visual angle metric, the triangular ratio metric, inner metric.

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Purpose Graphene, which has abundant availability in nature, is currently under research for its functional applications in the field of textiles. The sp2 Hybridized 1-atom-thick planar sheet has been under consideration for its unique electrical, mechanical and thermal properties, but there exists a void for aggregated data on the findings of other co-functional properties attained by the material using graphene oxide (GO) finish. This paper aims to define the techniques of extraction of GO, method of its application on textile material followed by detailed evaluation of the differential functional properties achieved. Design/methodology/approach The methodology used to explain the multiple functionalities of GO finish have been carried out by starting with the chemistry of graphene and the isolation of GO from graphite, followed by the techniques for its application on the textile along with the study on the induced functional properties that may aid to increase its potential applications. Findings It has been observed that with the aid of optimization of GO finish, the finish in lieu with the conductive potentialities may further provide with many essential properties such as hydrophobicity, ultraviolet protection and antibacterial property. Originality/value The field of research on GO finish is naive and except few properties, many functionalities are still unexplored that may enable its smooth production, handling and expanding its area of application. The agglomeration of scattered findings on the achievable functional properties of GO on various textiles has been achieved in this paper.

We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps.2010 Mathematics subject classification: primary 51M10; secondary 26A15, 30C20, 30C65, 30F45.

In this paper we prove a sharp distortion property of the Cassinian metric under M\"obius transformations of a punctured ball onto another punctured ball. The paper also deals with discussion on local convexity properties of the Cassinian metric balls in some specific subdomains of $\mathbb{R}^n$. Inclusion properties of the Cassinian metric balls with other hyperbolic-type metric balls are also investigated. In particular, several conjectures are also stated in response to sharpness of the inclusion relations.Comment: 16 pages, 2 figure