Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-17T14:33:49.549Z Has data issue: false hasContentIssue false
  • English
  • Français

Inequalities for symmetric means, symmetric harmonic means, and their applications

Published online by Cambridge University Press:  17 April 2009

Hsu-Tung Ku ,
Mei-Chin Ku  and
Xin-Min Zhang
Hsu-Tung Ku
Affiliation:
Department of Mathematics and StatisticsUniversity of Massachusetts at AmherstAmherst, MA 01003United States of America, e-mail: hku@math.umass.edu, meiku@math.umass.edu
Mei-Chin Ku
Affiliation:
Department of Mathematics and StatisticsUniversity of Massachusetts at AmherstAmherst, MA 01003United States of America, e-mail: hku@math.umass.edu, meiku@math.umass.edu
Xin-Min Zhang
Affiliation:
Department of Mathematics and StatisticsUniversity of South AlabamaMobile, AL 36688United States of America e-mail: zhang@mathstat.usouthal.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we establish a number of inequalities involving symmetric means and symmetric harmonic means. We then apply these new inequalities to obtain many geometric inequalities of isoperimetric type for plane polygons.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 409 - 420
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Beckenbach, E.F. and Bellman, R., Inequalities (Springer-Verlag, Berlin, Heidelberg, New York, 1965).CrossRefGoogle Scholar
[2]Brooks, R. and Wakesman, P., ‘The first eigenvalue of a scalene triangle’, Proc. Amer. Math. Soc. 100 (1987), 175–182..CrossRefGoogle Scholar
[3]Bullen, P.S., Mitrinović, D.S. and Vasić, P.M., Means and their inequalities (Reidel, Dordrecht, 1988).CrossRefGoogle Scholar
[4]Hardy, G., Littlewood, J.E. and Pólya, G., Inequalities (Cambridge University Press, Cambridge, New York, 1951).Google Scholar
[5]Kazarinoff, M.D., Geometric inequalities, New Math. Library (Random House, New York, 1961).CrossRefGoogle Scholar
[6]Ku, H.T., Ku, M.C. and Zhang, X.M., ‘Generalized power means and interpolating inequalities’, (preprint).Google Scholar
[7]Ku, H.T., Ku, M.C. and Zhang, X.M., ‘Analytic and geometric isoperimetric inequalities’, J. Geom. 53 (1995), 100–121.CrossRefGoogle Scholar
[8]Macnab, D.S., ‘Cyclic polygons and related questions’, Math. Gaz. 65 (1981), 22–28.CrossRefGoogle Scholar
[9]Mitrinović, D.S., Analytic inequalities (Springer-Verlag, Berlin, Heidelberg, New York, 1970).Google Scholar
[10]Mitrinović, D.S., Pečarić, J.E. and Fink, A.M., Classical and new inequalities in analysis (Kluwer Academic Publishers, Dordrecht, Boston, London, 1993).CrossRefGoogle Scholar
[11]Mitrinović, D.S., Pečarić, J.E. and Volenec, V., Recent advances in geometric inequalities (Kluwer Academic Publishers, Dordrecht, Boston, London, 1989).CrossRefGoogle Scholar
[12]Osserman, R., ‘The isoperimetric inequalities’, Bull. Amer. Math. Soc. 84 (1978), 1182–1238.CrossRefGoogle Scholar
[13]Pólya, G., Mathematics and plausible reasoning I, Induction and Analogy in Mathematics (Princeton University Press, Princeton, NJ, 1954.).Google Scholar
[14]Pólya, G., ‘On the eigenvalues of vibrating membrances’, London Math. Soc. 11 (1961), 414–433.Google Scholar
[15]Pólya, G. and Szegö, G., Isoperimetric inequalities in mathematical physics, Annals of Mathematics 27 (Princeton, NJ, 1951).Google Scholar
[16]Zhang, X.M., ‘Bonnesen-style inequalities and pseudo-perimeters for polygons’, J. Geom. (to appear).Google Scholar