Schur-convex functions and isoperimetric inequalities
HTML articles powered by AMS MathViewer
- by Xin-Min Zhang
- Proc. Amer. Math. Soc. 126 (1998), 461-470
- DOI: https://doi.org/10.1090/S0002-9939-98-04151-3
- PDF | Request permission
Abstract:
In this paper, we establish some analytic inequalities for Schur-convex functions that are made of solutions of a second order nonlinear differential equation. We apply these analytic inequalities to obtain some geometric inequalities.References
- Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 30, Springer-Verlag, Inc., New York, 1965. Second revised printing. MR 0192009, DOI 10.1007/978-3-662-35199-4
- William E. Boyce and Richard C. DiPrima, Elementary differential equations and boundary value problems, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0179403
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Nicholas D. Kazarinoff, Geometric inequalities, New Mathematical Library, vol. 4, Random House, New York-Toronto, 1961. MR 0130134, DOI 10.5948/UPO9780883859223
- Hsü Tung Ku, Mei Chin Ku, and Xin Min Zhang, Analytic and geometric isoperimetric inequalities, J. Geom. 53 (1995), no. 1-2, 100–121. MR 1337429, DOI 10.1007/BF01224044
- H. T. Ku, M. C. Ku and X. M. Zhang, Isoperimetric Inequalities on the Surfaces of Constant Curvature, (Preprint).
- D. S. Macnab, Cyclic polygons and related questions, Math. Gaz. 65 (1981), no. 431, 22–28. MR 619164, DOI 10.2307/3617929
- J. E. Marsden and A. J. Tromba, Vector Calculus, 2nd Ed., W. H. Freeman and Company, 1981.
- Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 552278
- D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686, DOI 10.1007/978-3-642-99970-3
- Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
- Robert Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1–29. MR 519520, DOI 10.2307/2320297
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- I. Schur, Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzunsber. Berlin. Math. Ges. 22(1923)9-20.
- A. Wayne Roberts and Dale E. Varberg, Convex functions, Pure and Applied Mathematics, Vol. 57, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0442824
- H. P. Wynn, An inequality for certain bivariate probability integrals, Biometrika 64 (1977), no. 2, 411–414. MR 501576, DOI 10.1093/biomet/64.2.411
- X.M.Zhang, Bonnesen-style Inequalities and Pseudo-perimeters for Polygons, Journal of Geometry, to appear.
- Xin-Min Zhang, A refinement of the discrete Wirtinger inequality, J. Math. Anal. Appl. 200 (1996), no. 3, 687–697. MR 1393109, DOI 10.1006/jmaa.1996.0232
Bibliographic Information
- Xin-Min Zhang
- Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688
- Email: zhang@mathstat.usouthal.edu
- Received by editor(s): September 8, 1995
- Received by editor(s) in revised form: August 11, 1996
- Communicated by: J. Marshall Ash
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 461-470
- MSC (1991): Primary 26B25, 26D05, 26D10, 52A40, 52A41, 52B60
- DOI: https://doi.org/10.1090/S0002-9939-98-04151-3
- MathSciNet review: 1423343