Skip to main content
SpringerLink
Log in

On some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions and their applications

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

In this paper, we establish two fractional integral equalities involving once and twice differential functions. Then, we apply such equalities to give some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions. Some applications to special means of positive real numbers are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avci, M., Kavurmaci, H., Özdemir, M.E.: New inequalities of Hermite-Hadamard type via \(s\)-convex functions in the second sense with applications. Appl. Math. Comput. 217, 5171–5176 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Bai, R., Qi, F., Xi, B.: Hermite-Hadamard type inequalities for the \(m\)- and \((\alpha, m)\)-logarithmically convex functions. Filomat 27, 1–7 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bessenyei, M.: The Hermite-Hadamard inequality in Beckenbach’s setting. J. Math. Anal. Appl. 364, 366–383 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Breckner, W.W.: Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen. Publ. Inst. Math. 23, 13–20 (1978)

    MATH  Google Scholar 

  5. Breckner, W.W., Orbn, B.: Continuity Properties of Rationally s-Convex Mappings with Values in an Ordered Topological Linear Space. Babeş-Bolyai University of Cluj-Napoca Publishing House, Cluj-Napoca (1978)

    Google Scholar 

  6. Cal, J., Carcamob, J., Escauriaza, L.: A general multidimensional Hermite-Hadamard type inequality. J. Math. Anal. Appl. 356, 659–663 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cristescu, G., Noor, M.A., Awan, M.U.: Bounds of the second degree cumulative frontier gaps of functions with generalized convexity. Carpathian J. Math 31, 173–180 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Dragomir, S.S.: Hermite-Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces. Linear Algebra Appl. 436, 1503–1515 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006)

    MATH  Google Scholar 

  10. Niculescu, C.P.: The Hermite-Hadamard inequality for log-convex functions. Nonlinear Anal. Theory Methods Appl. 75, 662–669 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Noor, M.A., Noor, K.I., Awan, M.U., Khan, S.: Fractional Hermite-Hadamard Inequalities for some New Classes of Godunova-Levin functions. Appl. Math. Inf. Sci. 8, 2865–2872 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Noor, M.A., Cristescu, G., Awan, M.U.: Generalized fractional Hermite-Hadamard inequalities for twice differentiable s-convex functions. Filomat 29, 807–815 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Özdemir, M.E., Avci, M., Kavurmaci, H.: Hermite-Hadamard-type inequalities via \((\alpha, m)\)-convexity. Comput. Math. Appl. 61, 2614–2620 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pearce, C.E.M.: Pec̆arić, J.: Inequalities for differentiable mappings with application to special means and quadrature formula. Appl. Math. Lett. 13, 51–55 (2000)

    Article  MathSciNet  Google Scholar 

  15. Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013)

    Article  MATH  Google Scholar 

  16. Tseng, K., Hwang, S., Hsu, K.: Hadamard-type and Bullen-type inequalities for Lipschitzian functions and their applications. Comput. Math. Appl. 64, 651–660 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tunç, M.: Some Hadamard-like inequalities via convex and \(s\)-convex functions and their applications for special means. Mediterr. J. Math. 11, 1047–1059 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, J., Deng, J., Fečkan, M.: Hermite-Hadamard type inequalities for \(r\)-convex functions via Riemann-Liouville fractional integrals. Ukr. Math. J. 65, 193–211 (2013)

    Article  MATH  Google Scholar 

  19. Wang, J., Li, X., Fečkan, M., Zhou, Y.: Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity. Appl. Anal. Int. J. 92, 2241–2253 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too.

Author information

Authors and Affiliations

  1. College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030031, Shanxi, People’s Republic of China

    Zhuoyan Gao

  2. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, People’s Republic of China

    Mengmeng Li & JinRong Wang

Authors
  1. Zhuoyan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mengmeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Authors’s contributions

This work was carried out in collaboration between all authors. GZY, LMM and WJR proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.

Additional information

This work is supported by National Natural Science Foundation of China (11201091).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, Z., Li, M. & Wang, J. On some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions and their applications. Bol. Soc. Mat. Mex. 23, 691–711 (2017). https://doi.org/10.1007/s40590-016-0087-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40590-016-0087-9

Keywords

Mathematics Subject Classification

Search

Navigation