Journal of Applied Mathematics

Refinements of Generalized Aczél's Inequality and Bellman's Inequality and Their Applications

Jing-Feng Tian and Shu-Yan Wang

Full-text: Open access

Abstract

We give some refinements of generalized Aczél's inequality and Bellman's inequality proposed by Tian. As applications, some refinements of integral type of generalized Aczél's inequality and Bellman's inequality are given.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 645263, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807846

Digital Object Identifier
doi:10.1155/2013/645263

Mathematical Reviews number (MathSciNet)
MR3032193

Zentralblatt MATH identifier
1266.49020

Citation

Tian, Jing-Feng; Wang, Shu-Yan. Refinements of Generalized Aczél's Inequality and Bellman's Inequality and Their Applications. J. Appl. Math. 2013 (2013), Article ID 645263, 6 pages. doi:10.1155/2013/645263. https://projecteuclid.org/euclid.jam/1394807846


Export citation

References

  • J. Aczél, “Some general methods in the theory of functional equations in one variable. New applications of functional equations,” Uspekhi Matematicheskikh Nauk, vol. 11, no. 3, pp. 3–68, 1956 (Russian).
  • E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin, Germany, 1983.
  • J. L. Díaz-Barrero, M. Grau-Sánchez, and P. G. Popescu, “Refinements of Aczél, Popoviciu and Bellman's inequalities,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2356–2359, 2008.
  • G. Farid, J. Pečarić, and A. Ur Rehman, “On refinements of Aczél, Popoviciu, Bellman's inequalities and related results,” Journal of Inequalities and Applications, vol. 2010, Article ID 579567, 17 pages, 2010.
  • G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 1952.
  • D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61, Kluwer Academic, Dordrecht, The Netherlands, 1993.
  • Y. Ouyang and R. Mesiar, “On the Chebyshev type inequality for seminormed fuzzy integral,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1810–1815, 2009.
  • T. Popoviciu, “On an inequality,” Gazeta Matematica şi Fizica A, vol. 11, no. 64, pp. 451–461, 1959 (Romanian).
  • J. Tian, “Inequalities and mathematical properties of uncertain variables,” Fuzzy Optimization and Decision Making, vol. 10, no. 4, pp. 357–368, 2011.
  • J. Tian, “Reversed version of a generalized Aczél's inequality and its application,” Journal of Inequalities and Applications, vol. 2012, article 202, 2012.
  • J. Tian, “Reversed version of a generalized sharp Hölder's inequality and its applications,” Information Sciences, vol. 201, pp. 61–69, 2012.
  • J. Tian, “Extension of Hu Ke's inequality and its applications,” Journal of Inequalities and Applications, vol. 2011, article 77, 2011.
  • J. Tian, “Property of a Hölder-type inequality and its application,” Mathematical Inequalities & Applications. In press.
  • J. Tian and X. M. Hu, “A new reversed version of a generalized sharp Hölder's inequality čommentComment on ref. [15?]: Please update the information of these references[14,15?], if possible.and its applications,” Abstract and Applied Analysis. In press.
  • P. M. Vasić and J. E. Pečarić, “On the Hölder and some related inequalities,” Mathematica, vol. 25, no. 1, pp. 95–103, 1982.
  • P. M. Vasić and J. E. Pečarić, “On the Jensen inequality for monotone functions,” Analele Universitatii din Timişoara, vol. 17, no. 1, pp. 95–104, 1979.
  • S. Vong, “On a generalization of Aczél's inequality,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1301–1307, 2011.
  • S. Wu and L. Debnath, “Generalizations of Aczél's inequality and Popoviciu's inequality,” Indian Journal of Pure and Applied Mathematics, vol. 36, no. 2, pp. 49–62, 2005.
  • W. Yang, “Refinements of generalized Aczél-Popoviciu's inequality and Bellman's inequality,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3570–3577, 2010.
  • R. Bellman, “On an inequality concerning an indefinite form,” The American Mathematical Monthly, vol. 63, pp. 108–109, 1956.