Advertisement

Grüss-Type Inequalities by Means of Generalized Fractional Integrals

  • J. Vanterler da C. Sousa
  • D. S. OliveiraEmail author
  • E. Capelas de Oliveira
Article
  • 45 Downloads

Abstract

We use a recently proposed fractional integral to establish a generalization of Grüss-type integral inequalities. We prove two theorems about these inequalities and enunciate and prove other inequalities associated with this fractional operator.

Keywords

Fractional integral Generalization inequalities of Grüss-type 

Notes

Acknowledgements

The authors are indebted to Dr. J. Emílio Maiorino for useful discussions about the theme. We would like to thank the referees for their valuable comments that improved this manuscript.

References

  1. Akin, E., Aslıyüce, S., Güvenilir, A.F., Kaymakçalan, B.: Discrete Grüss type inequality on fractional calculus. J. Inequal. Appl. 2015(1), 174 (2015)CrossRefzbMATHGoogle Scholar
  2. Almeida, R.: A Gronwall inequality for a general Caputo fractional operator. Math. Inequal. Appl. 20(4), 1089–1105 (2017)MathSciNetzbMATHGoogle Scholar
  3. Beesack, P.R.: Hardy’s inequality and its extensions. Pacific J. Math. 11(1), 39–61 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bougoffa, L.: On Minkowski and Hardy integral inequalities. J. Inequal. Pure Appl. Math. 7(2), 1–3 (2006)MathSciNetzbMATHGoogle Scholar
  5. Camargo, R.F., Oliveira, E.C.: Fractional Calculus (in Portuguese), Editora Livraria da Física, São Paulo (2015)Google Scholar
  6. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015)Google Scholar
  7. Chen, H., Katugampola, U.N.: Hermite-Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals. J. Math. Anal. and Appl. 446(2), 1274–1291 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chinchane, V.L., Pachpatte, D.B.: On some new Gruss-type inequality using Hadamard fractional integral operator. J. Frac. Calc. Appl. 5(12), 1–10 (2014)Google Scholar
  9. Chinchane, V.L., Pachpatte, D.B.: New fractional inequalities involving Saigo fractional integral operator. Math. Sci. Lett. 3(3), 133–139 (2014)CrossRefzbMATHGoogle Scholar
  10. Dahmani, Z.: On Minkowski and Hermite-Hadamard integral inequalities via fractional integration. Ann. Funct. Anal. 1(1), 51–58 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Dahmani, Z., Tabharit, L., Taf, S.: New generalisations of Gruss inequality using Riemann–Liouville fractional integrals. Bull. Math. Anal. Appl. 2(3), 93–99 (2010)MathSciNetzbMATHGoogle Scholar
  12. Dragomir, S.S.: A Grüss type discrete inequality in inner product spaces and applications. J. Math. Anal. and Appl. 250, 494–511 (2000)Google Scholar
  13. Grüss, G.: Über das maximum des absoluten betrages von \(\dfrac{1}{b-a} \int _{a}^{b}f(x) g(x)dx-\dfrac{1}{(b-a)^2}\) \(\int _{a}^{b}f(x) dx \int _{a}^{b}g(x) dx \). Math. Z. 39(1), 215–226 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hutník, O.: On Hadamard type inequalities for generalized weighted quasi-arithmetic means. J. Inequal. Pure Appl. Math. 7(3), 1–10 (2006)MathSciNetzbMATHGoogle Scholar
  15. Hutník, O.: Some integral inequalities of Hölder and Minkowski type. Colloq. Math. 108, 247–261 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. İşcan, İ.: Ostrowski type inequalities for harmonically s-convex functions. Konuralp J. Math. 3(1), 63–74 (2015)MathSciNetzbMATHGoogle Scholar
  17. İşcan, İ.: Hermite–Hadamard-Fejér type inequalities for convex functions via fractional integrals. Stud. Univ. Babeş-Bolyai Math. 60(3), 355–366 (2015)zbMATHGoogle Scholar
  18. Katugampola, U.N.: New fractional integral unifying six existing fractional integrals, pp. 6. (2016) arXiv:1612.08596 (eprint)
  19. Katugampola, U.N.: New approach to a generalized fractional integral. App. Math. Comput. 218, 860–865 (2011).  https://doi.org/10.1016/j.amc.2011.03.062 MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of the Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  21. Kiliçman, A., Saleh, W.: On some inequalities for generalized s-convex functions and applications on fractal sets. J. Nonlinear Sci. Appl. 10, 583–594 (2017)MathSciNetCrossRefGoogle Scholar
  22. Kreyszig, E.: Introductory Functional Analysis with Applications, vol. 1. Wiley, New York (1989)zbMATHGoogle Scholar
  23. Lin, S.: Generalized Gronwall inequalities and their applications to fractional differential equations. J. Ineq. Appl. 2013, 2013–549 (2013)MathSciNetGoogle Scholar
  24. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 87–92 (2015)Google Scholar
  25. Minculete, N., Ciurdariu, L.: A generalized form of Grüss type inequality and other integral inequalities. J. Inequal. Appl. 2014(1), 119 (2014)CrossRefzbMATHGoogle Scholar
  26. Oguntuase, J.A., Imoru, C.O.: New generalizations of Hardy’s integral inequality. J. Math. Anal. and Appl. 241(1), 73–82 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Sarikaya, M.Z., Budak, H.U.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145, 1527–1538 (2017)MathSciNetzbMATHGoogle Scholar
  28. Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. and Comp. Model. 57(9), 2403–2407 (2013)CrossRefzbMATHGoogle Scholar
  29. Set, E.., Özdemir, M., Dragomir, S.: On the Hermite-Hadamard inequality and other integral inequalities involving two functions. J. Inequal. Appl. 2010(1), 9 (2010)Google Scholar
  30. Sousa, V.C.J., Capelas de Oliveira, E., Magna, L.A.: Fractional calculus and the ESR test. AIMS Math. 2(4), 692–705 (2017)Google Scholar
  31. Sousa, V.C.J., Capelas de Oliveira, E.: The Minkowski’s inequality by means of a generalized fractional integral. Accepted for publication AIMS Math. 3(1), 131–147 (2018)Google Scholar
  32. Sousa, V.C.J., dos Santos, Magun N.N., Magna, L.A., Capelas de Oliveira, E.: Validation of a fractional model for erythrocyte sedimentation rate. Comput. Appl. Math. 37, 6903–6919 (2018)Google Scholar
  33. Steele, J.M.: The Cauchy–Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  34. Szeligowska, W., Kaluszka, M.: On Jensen’s inequality for generalized Choquet integral with an application to risk aversion. arXiv:1609.00554 (2016) (eprint)
  35. Taf, S., Brahim, K.: Some new results using Hadamard fractional integral. Int. J. Nonlinear Anal. Appl 2(2), 24–42 (2015)zbMATHGoogle Scholar
  36. Vanterler da C. Sousa, J., Capelas de Oliveira, E.: A new truncated \(M\)-fractional derivative type unifying some fractional derivative types with classical properties. Int. J. Anal. Appl. 16(1), 83–96 (2018)zbMATHGoogle Scholar
  37. Yang, X.-J.: Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm. Sci. 21, 1161–1171 (2017)CrossRefGoogle Scholar
  38. Yang, X.-J., Tenreiro Machado, J.A.: A new fractional operator of variable order: application in the description of anomalous diffusion. Phys. A: Stat. Mech. Appl. 481, 276–283 (2017)MathSciNetCrossRefGoogle Scholar
  39. Yang, X.-J., Srivastava, H.M., Tenreiro Machado, J.A.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20, 753–756 (2017)CrossRefGoogle Scholar
  40. Zhu, C., Yang, W., Zhao, Q.: Some new fractional \(q\)-integral Gruss-type inequalities and other inequalities. J. Inequa. and Appl. 2012(1), 15 (2012)Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  • J. Vanterler da C. Sousa
    • 1
  • D. S. Oliveira
    • 2
    Email author
  • E. Capelas de Oliveira
    • 1
  1. 1.Department of Applied MathematicsImecc–UnicampCampinasBrazil
  2. 2.Coordination of Civil EngineeringTechnological Federal University of ParanáGuarapuavaBrazil

Personalised recommendations