Open Access

One Method for Proving Inequalities by Computer

Journal of Inequalities and Applications20072007:078691

https://doi.org/10.1155/2007/78691

Received: 31 August 2006

Accepted: 31 October 2006

Published: 9 January 2007

Abstract

We consider a numerical method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple computer program.

[12345678910111213141516171819202122232425262728]

Authors’ Affiliations

(1)
Faculty of Electrical Engineering, University of Belgrade

References

  1. Malešević BJ: Some inequalities for Kurepa's function. Journal of Inequalities in Pure and Applied Mathematics 2004,5(4):5 pages.Google Scholar
  2. Abramowitz M, Stegun IA (Eds): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, USA National Bureau of Standards Applied Mathematics. Volume 55. 10th edition. Wiley-Interscience, New York, NY, USA; 1972.Google Scholar
  3. Geddes KO: A Package for Numerical Approximation. The Maple Technical Newsletter, 10, Fall 1993, 28–36, http://www.adeptscience.co.uk/maplearticles/f231.html.
  4. Fike CT: Computer Evaluation of Mathematical Functions. Prentice-Hall, Englewood Cliffs, NJ, USA; 1968:xii+227.MATHGoogle Scholar
  5. de Dinechin F, Lauter C, Melquiond G: Assisted verification of elementary functions. In Research Report 5683. INRIA, Montbonnot Saint Ismier, France; 2005. 17 pages, http://www.inria.fr/rrrt/rr-5683.html 17 pages,Google Scholar
  6. Brisebarre N, Muller J-M, Tisserand A: Computing machine-efficient polynomial approximation. ACM Transactions on Mathematical Software 2006,32(2):236–256. 10.1145/1141885.1141890MathSciNetView ArticleGoogle Scholar
  7. Hart JF: Computer Approximations. Krieger, Melbourne, Fla, USA; 1978.Google Scholar
  8. Kurepa D: Left factorial function in complex domain. Mathematica Balkanica 1973, 3: 297–307.MathSciNetMATHGoogle Scholar
  9. Malešević BJ: Some considerations in connection with Kurepa's function. Univerzitet u Beogradu, Publikacije Elektrotehničkog Fakulteta, Serija Matematika 2003, 14: 26–36.MATHGoogle Scholar
  10. Slavić DV: On the left factorial function of the complex argument. Mathematica Balkanica 1973, 3: 472–477.MathSciNetMATHGoogle Scholar
  11. Petojević A: New formulae forfunction. Novi Sad Journal of Mathematics 2005,35(2):123–132.MathSciNetMATHGoogle Scholar
  12. Mitrinović DS, Vasić PM: Analytic Inequalities, Die Grundlehren der Mathematischen Wisenschaften, Band 1965. Springer, New York, NY, USA; 1970:xii+400.Google Scholar
  13. Fink AM: Two inequalities. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 1995, 6: 48–49.MathSciNetMATHGoogle Scholar
  14. Malešević BJ: Application of-method on Shafer-Fink's inequality. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 1997, 8: 103–105.MATHGoogle Scholar
  15. Zhu L: On Shafer-Fink inequalities. Mathematical Inequalities & Applications 2005,8(4):571–574.MathSciNetView ArticleMATHGoogle Scholar
  16. Malešević BJ: An application of-method on inequalities of Shafer-Fink's type. preprint, http://arxiv.org/abs/math.CA/0608787 preprint,
  17. Golub GH, Smith LB: Chebyshev approximation of continuous functions by a Chebyshev system of functions. Communications of the ACM 1971,14(11):737–746. 10.1145/362854.362890View ArticleGoogle Scholar
  18. Elbert Á, Laforgia A: On some properties of the gamma function. Proceedings of the American Mathematical Society 2000,128(9):2667–2673. 10.1090/S0002-9939-00-05520-9MathSciNetView ArticleMATHGoogle Scholar
  19. Cerone P: On applications of the integral of products of functions and its bounds. RGMIA: Journal of Inequalities in Pure and Applied Mathematics 2003.,6(4):Google Scholar
  20. Guo B-N, Qiao B-M, Qi F, Li W: On new proofs of Wilker's inequalities involving trigonometric functions. Mathematical Inequalities & Applications 2003,6(1):19–22.MathSciNetView ArticleMATHGoogle Scholar
  21. Alzer H: On Ramanujan's double inequality for the gamma function. The Bulletin of the London Mathematical Society 2003,35(5):601–607. 10.1112/S0024609303002261MathSciNetView ArticleMATHGoogle Scholar
  22. Batir N: Some new inequalities for gamma and polygamma functions. Journal of Inequalities in Pure and Applied Mathematics 2005,6(4):9 pages.MathSciNetGoogle Scholar
  23. Chen C-P, Qi F: The best bounds in Wallis' inequality. Proceedings of the American Mathematical Society 2005,133(2):397–401. 10.1090/S0002-9939-04-07499-4MathSciNetView ArticleMATHGoogle Scholar
  24. Zhu L: Sharpening Jordan's inequality and the Yang Le inequality. Applied Mathematics Letters 2006,19(3):240–243. 10.1016/j.aml.2005.06.004MathSciNetView ArticleMATHGoogle Scholar
  25. Wua S, Debnath L: A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality. Applied Mathematics Letters 2006,19(12):1378–1384. 10.1016/j.aml.2006.02.005MathSciNetView ArticleGoogle Scholar
  26. Qiu S-L, Vamanamurthy MK, Vuorinen M: Some inequalities for the Hersch-Pfluger distortion function. Journal of Inequalities and Applications 1999,4(2):115–139. 10.1155/S1025583499000326MathSciNetMATHGoogle Scholar
  27. Dragomir SS, Agarwal RP, Barnett NS: Inequalities for beta and gamma functions via some classical and new integral inequalities. Journal of Inequalities and Applications 2000,5(2):103–165. 10.1155/S1025583400000084MathSciNetMATHGoogle Scholar
  28. Laforgia A, Natalini P: Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions. Journal of Inequalities and Applications 2006, 2006: 8 pages.Google Scholar

Copyright

© Branko J. Malešević 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.