Ceived no external funding. Institutional Assessment Board Statement: Not applicable. Informed Consent Statement: Not applicable. Information Availability Statement: Not applicable. Acknowledgments: The authors would like to thank anonymous reviewers as well as the editor for their comments and suggestions to enhance this paper. Conflicts of Interest: The authors declare no conflict of interest.SAppendix A. Existing Index S Assuming that ij ji 0 for all i = j, the index S , which represents the degree of deviation from S, is expressed as follows: S exactly where IS with = =( 1) ( ) I , 2 – 1 Saij 1 = aij bij ( 1) i = j aij = ij , bij =- 1,ij ji .ij ,i=jAppendix B. Existing Index PS Assuming that ij i j 0 for all (i, j) E, the index PS , which represents the degree of deviation from PS, is expressed as follows: PS =( 1) ( ) I , 2 – 1 PSSymmetry 2021, 13,10 ofwhere IPS with = (i,j) E cij 1 = cij dij ( 1) (i,j)E cij = ij , dij =- 1,ij i j .ij ,Note that is and IPS are the energy divergence among the two conditional distributions, as well as the value at = 0 is taken to become the limit as 0.
Citation: Mocanu, M. Functional Inequalities for Metric-Preserving Functions with Respect to Intrinsic Metrics of Hyperbolic Sort. Symmetry 2021, 13, 2072. https:// doi.org/10.3390/sym13112072 Academic Editors: Wlodzimierz Fechner and Jacek Chudziak Received: 28 September 2021 Accepted: 24 October 2021 Published: 2 NovemberPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Metric-preserving functions happen to be Hydroxyflutamide MedChemExpress studied normally topology from a theoretical point of view and have applications in fixed point theory [1,2], as well as in metric geometry d to construct new metrics from identified metrics, as the metrics 1d , log(1 d) and also the , (0, 1) connected to every single metric d [3]. The theory of metric-preserving -snowlake d functions, that may be traced back to Wilson and Blumenthal, has been developed by Bors , Dobos, Piotrowski, Vallin [6] and other people, being not too long ago generalized to semimetric spaces and quasimetric spaces [10] (see also [113]). As we will show beneath, there’s a strong connection between metric-preserving functions and subadditive functions. The theory of subadditive function is well-developed [14,15], the functional inequality corresponding to subadditivity getting viewed as a all-natural counterpart of Cauchy functional equation [16,17]. Provided a function f : [0, ) [0, ), it can be mentioned that f is metric-preserving if for every single metric space ( X, d) the function f d is also a metric on X, i.e., f PK 11195 Autophagy transfers each metric to a metric The function f : [0, ) [0, ) is known as amenable if f -1 (0) = 0. If there exists some metric space ( X, d) such that the function f d is also a metric on X, then f : [0, ) [0, ) is amenable. The symmetry axiom of a metric is naturally happy by f d anytime d can be a metric. Given f amenable, f is metric-preserving if and only if f d satisfies triangle inequality whenever d is really a metric. Every single on the following properties is recognized to be a adequate situation for an amenable function to become metric-preserving [10,11]: 1. 2. three. f is concave; f is nondecreasing and subadditive; f is tightly bounded (that is definitely, there exists a 0 such that f ( x ) [ a, 2a] for each and every x 0).f (t)Copyright: 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is definitely an open access post distributed under the terms and situations in the Inventive Commons Attributi.
