Open Access

Upper bounds for the eigenvalues of differential equations

Journal of Inequalities and Applications20062006:48606

DOI: 10.1155/JIA/2006/48606

Received: 10 February 2004

Accepted: 4 May 2004

Published: 3 January 2006


We establish upper bounds for the eigenvalues of second-order and fourth-order differential equations. The inequalities are obtained via rearrangements of higher degree.


Authors’ Affiliations

Department of Mathematics and Statistics, Sultan Qaboos University


  1. Bandle C: Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics. Volume 7. Pitman, Massachusetts; 1980:x+228.Google Scholar
  2. Bandle C: Extremal problems for eigenvalues of the Sturm-Liouville type. In General Inequalities, 5 (Oberwolfach, 1986), Internat. Schriftenreihe Numer. Math.. Volume 80. Birkhäuser, Basel; 1987:319–336.Google Scholar
  3. Banks DO: Bounds for the eigenvalues of nonhomogeneous hinged vibrating rods. Journal of Mathematics Mechanics 1967, 16: 949–966.MATHMathSciNetGoogle Scholar
  4. Barnes DC: Rearrangements of functions and lower bounds for eigenvalues of differential equations. Applicable Analysis. An International Journal 1982,13(4):237–248.MATHMathSciNetView ArticleGoogle Scholar
  5. Barnes DC: Extremal problems for eigenvalues with applications to buckling, vibration and sloshing. SIAM Journal on Mathematical Analysis 1985,16(2):341–357. 10.1137/0516025MATHMathSciNetView ArticleGoogle Scholar
  6. Cochran JA: The Analysis of Linear Integral Equations. McGraw-Hill, New York; 1972:xi+370.MATHGoogle Scholar
  7. Cox SJ, McCarthy CM: The shape of the tallest column. SIAM Journal on Mathematical Analysis 1998,29(3):547–554. 10.1137/S0036141097314537MATHMathSciNetView ArticleGoogle Scholar
  8. Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge; 1934.MATHGoogle Scholar
  9. Karaa S: Sharp estimates for the eigenvalues of some differential equations. SIAM Journal on Mathematical Analysis 1998,29(5):1279–1300. 10.1137/S0036141096307849MATHMathSciNetView ArticleGoogle Scholar
  10. Karaa S: Inequalities for eigenvalue functionals. Journal of Inequalities and Applications 1999,4(2):175–181. 10.1155/S1025583499000351MATHMathSciNetGoogle Scholar
  11. Schwarz B: On the extrema of the frequencies of nonhomogeneous strings with equimeasurable density. Journal of Mathematics Mechanics 1961, 10: 401–422.MATHMathSciNetGoogle Scholar


© Hindawi Publishing Corporation. 2006

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