- Research Article
- Open access
- Published:
One Method for Proving Inequalities by Computer
Journal of Inequalities and Applications volume 2007, Article number: 078691 (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.
References
Malešević BJ: Some inequalities for Kurepa's function. Journal of Inequalities in Pure and Applied Mathematics 2004,5(4):5 pages.
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.
Geddes KO: A Package for Numerical Approximation. The Maple Technical Newsletter, 10, Fall 1993, 28–36, http://www.adeptscience.co.uk/maplearticles/f231.html.
Fike CT: Computer Evaluation of Mathematical Functions. Prentice-Hall, Englewood Cliffs, NJ, USA; 1968:xii+227.
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,
Brisebarre N, Muller J-M, Tisserand A: Computing machine-efficient polynomial approximation. ACM Transactions on Mathematical Software 2006,32(2):236–256. 10.1145/1141885.1141890
Hart JF: Computer Approximations. Krieger, Melbourne, Fla, USA; 1978.
Kurepa D: Left factorial function in complex domain. Mathematica Balkanica 1973, 3: 297–307.
Malešević BJ: Some considerations in connection with Kurepa's function. Univerzitet u Beogradu, Publikacije Elektrotehničkog Fakulteta, Serija Matematika 2003, 14: 26–36.
Slavić DV: On the left factorial function of the complex argument. Mathematica Balkanica 1973, 3: 472–477.
Petojević A: New formulae forfunction. Novi Sad Journal of Mathematics 2005,35(2):123–132.
Mitrinović DS, Vasić PM: Analytic Inequalities, Die Grundlehren der Mathematischen Wisenschaften, Band 1965. Springer, New York, NY, USA; 1970:xii+400.
Fink AM: Two inequalities. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 1995, 6: 48–49.
Malešević BJ: Application of-method on Shafer-Fink's inequality. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 1997, 8: 103–105.
Zhu L: On Shafer-Fink inequalities. Mathematical Inequalities & Applications 2005,8(4):571–574.
Malešević BJ: An application of-method on inequalities of Shafer-Fink's type. preprint, http://arxiv.org/abs/math.CA/0608787 preprint,
Golub GH, Smith LB: Chebyshev approximation of continuous functions by a Chebyshev system of functions. Communications of the ACM 1971,14(11):737–746. 10.1145/362854.362890
Elbert Á, Laforgia A: On some properties of the gamma function. Proceedings of the American Mathematical Society 2000,128(9):2667–2673. 10.1090/S0002-9939-00-05520-9
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):
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.
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/S0024609303002261
Batir N: Some new inequalities for gamma and polygamma functions. Journal of Inequalities in Pure and Applied Mathematics 2005,6(4):9 pages.
Chen C-P, Qi F: The best bounds in Wallis' inequality. Proceedings of the American Mathematical Society 2005,133(2):397–401. 10.1090/S0002-9939-04-07499-4
Zhu L: Sharpening Jordan's inequality and the Yang Le inequality. Applied Mathematics Letters 2006,19(3):240–243. 10.1016/j.aml.2005.06.004
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.2006.02.005
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/S1025583499000326
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/S1025583400000084
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Malešević, B.J. One Method for Proving Inequalities by Computer. J Inequal Appl 2007, 078691 (2007). https://doi.org/10.1155/2007/78691
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2007/78691