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


Two new fractal measures and are constructed from Minkowski contents and . The properties of these two new measures are studied. We show that the fractal dimensions Dim and can be derived from and , respectively. Moreover, some inequalities about the dimension of product sets and product measures are obtained.


Authors’ Affiliations

Department of Mathematics, Georgetown University
School of Computer science and Engineering, South China University of Technology


  1. Falconer K: Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Chichester, UK; 1990:xxii+288.MATHGoogle Scholar
  2. Falconer KJ: The Geometry of Fractal Sets, Cambridge Tracts in Mathematics. Volume 85. Cambridge University Press, Cambridge, UK; 1986:xiv+162.Google Scholar
  3. Mattila P: Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics. Volume 44. Cambridge University Press, Cambridge, UK; 1995:xii+343.View ArticleGoogle Scholar
  4. Taylor SJ, Tricot C: Packing measure, and its evaluation for a Brownian path. Transactions of the American Mathematical Society 1985,288(2):679–699. 10.1090/S0002-9947-1985-0776398-8MathSciNetView ArticleMATHGoogle Scholar
  5. Tricot C Jr.: Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society 1982,91(1):57–74. 10.1017/S0305004100059119MathSciNetView ArticleMATHGoogle Scholar
  6. Besicovitch AS, Moran PAP: The measure of product and cylinder sets. Journal of the London Mathematical Society. Second Series 1945, 20: 110–120. 10.1112/jlms/s1-20.2.110MathSciNetMATHGoogle Scholar
  7. Xiao Y: Packing dimension, Hausdorff dimension and Cartesian product sets. Mathematical Proceedings of the Cambridge Philosophical Society 1996,120(3):535–546. 10.1017/S030500410007506XMathSciNetView ArticleMATHGoogle Scholar
  8. Haase H: On the dimension of product measures. Mathematika 1990,37(2):316–323. 10.1112/S0025579300013024MathSciNetView ArticleMATHGoogle Scholar
  9. Stein EM, Shakarchi R: Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ, USA; 2005:xx+402.Google Scholar
  10. Bishop CJ, Peres Y: Packing dimension and Cartesian products. Transactions of the American Mathematical Society 1996,348(11):4433–4445. 10.1090/S0002-9947-96-01750-3MathSciNetView ArticleMATHGoogle Scholar


© 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.