Open Access

New Inequalities on Fractal Analysis and Their Applications

Journal of Inequalities and Applications20072007:026249

DOI: 10.1155/2007/26249

Received: 26 September 2006

Accepted: 23 November 2006

Published: 21 February 2007

Abstract

Two new fractal measures https://static-content.springer.com/image/art%3A10.1155%2F2007%2F26249/MediaObjects/13660_2006_Article_1686_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F26249/MediaObjects/13660_2006_Article_1686_IEq2_HTML.gif are constructed from Minkowski contents https://static-content.springer.com/image/art%3A10.1155%2F2007%2F26249/MediaObjects/13660_2006_Article_1686_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F26249/MediaObjects/13660_2006_Article_1686_IEq4_HTML.gif . The properties of these two new measures are studied. We show that the fractal dimensions Dim and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F26249/MediaObjects/13660_2006_Article_1686_IEq5_HTML.gif can be derived from https://static-content.springer.com/image/art%3A10.1155%2F2007%2F26249/MediaObjects/13660_2006_Article_1686_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F26249/MediaObjects/13660_2006_Article_1686_IEq7_HTML.gif , respectively. Moreover, some inequalities about the dimension of product sets and product measures are obtained.

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Authors’ Affiliations

(1)
Department of Mathematics, Georgetown University
(2)
School of Computer science and Engineering, South China University of Technology

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Copyright

© D.-C. Chang and Y. Xu 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.