Hermite-Hadamard Type Inequalities for Quasi-Convex Functions via Katugampola Fractional Integrals

Set, Erhan; Mumcu, Ilker

doi:10.28924/2291-8639-16-2018-605

Research Article

Hermite-Hadamard Type Inequalities for Quasi-Convex Functions via Katugampola Fractional Integrals

Erhan Set¹*
Ilker Mumcu

Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey

* Corresponding author

Abstract

The paper deals with quasi-convex functions, Katugampola fractional integrals and Hermite-Hadamard type integral inequalities. The main idea of this paper is to present new Hermite-Hadamard type inequalities for quasi-convex functions using Katugampola fractional integrals, Hölder inequality and the identities in the literature.

How to Cite

E. Set, I. Mumcu, Hermite-Hadamard Type Inequalities for Quasi-Convex Functions via Katugampola Fractional Integrals, Int. J. Anal. Appl. 16 (2018), 605–613. https://doi.org/10.28924/2291-8639-16-2018-605

BibTeX

@article{Set2018,
  author  = {Set, Erhan and Mumcu, Ilker},
  title   = {Hermite-Hadamard Type Inequalities for Quasi-Convex Functions via Katugampola Fractional Integrals},
  journal = {Int. J. Anal. Appl.},
  volume  = {16},
  year    = {2018},
  pages   = {605--613},
  doi     = {10.28924/2291-8639-16-2018-605}
}

References

[1] M.U. Awan, M.A. Noor, M.V. Mihai and K.I. Noor, Fractional Hermite-Hadamard inequalities for differentiables- Godunova-Levin functions, Filomat, 30(12) (2016), 3235C-3241.
[2] H. Chen, U.N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446, 1274-1291.
[3] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Mono- graphs, Victoria University, 2000.
[4] D.A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Ann. Univ. Craiova, Math. Comp. Sci. Ser., 34 (2007), 82-87.
[5] U.N. Katugampola, New approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6(4), (2014), 1-15.
[6] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Else- vier; (2006).
[7] M. A. Noor, K. I. Noor, and M. U. Awan, Generalized convexity and integral inequalities, Appl. Math. Inform. Sci., 9(1) (2015),233-243.
[8] M. E.Özdemir, C ¸. Yildiz, The Hadamards inequality for quasi-convex functions via fractional integrals, Annals of the University of Craiova, Ann. Univ. Craiova, Math. Comp. Sci. Series, 40(2) (2013), 167-173.
[9] A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integral and series. In: Elementary Functions, vol. 1. Nauka, Moscow; (1981).
[10] E.D. Rainville, Special Functions, The Mcmillan Company, New York, 1960.
[11] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Ba ¸sak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403-2407.
[12] E. Set, M.Z. Sarikaya, M.E.Özdemir, H. Yildirim, The Hermite-Hadamards inequality for some convex functions via fractional integrals and related results, J. Appl. Math. Stat. Inf., 10(2) (2014), 69-83.
[13] E. Set,Ë™I.Ë™I ¸scan, F. Zehir, On some new inequalities of Hermite-Hadamard type involving harmonically convex functions via fractional integrals, Konuralp J. Math., 3(1) (2015), 42-55.
[14] E. Set, A.O. Akdemir,Ë™I. Mumcu, Hermite-Hadamard's Inequality and its Extensions for Conformable Fractional Integrals of Any Order α > 0, ResearchGate, https://www.researchgate.net/publication/303382221.
[15] E. Set, B. C ¸elik, Fractional Hermite-Hadamard type inequalities for quasi-convex functions, Ordu Univ. J. Sci. Tech., 6(1) (2016), 137-149.
[16] E. Set, A.O. Akdemir, B. C ¸elik, On Generalization of Fejer Type Inequalities via fractional integral operator, ResearchGate, https://www.researchgate.net/publication/311452467.
[17] E. Set, A.O. Akdemir,˙I. Mumcu, Ostrowski type inequalities involving special functions via conformable fractional integrals, J. Adv. Math. Stud., 10(3) (2017), 386-395.
[18] E. Set, M.A. Noor, M.U. Awan, A. Gözpinar, Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Ineq. Appl., 2017(169) (2017), 1-10.
[19] E. Set,J. Choi, A. Gözpinar, Hermite-Hadamard type inequalities for the generalized k-fractional integral operators, J. Ineq. Appl., 2017 (2017), Art. ID 206.
[20] E. Set, M.E.Özdemir, M.Z. Sarikaya, Inequalities of Hermite-Hadamard type for functions whose derivatives absolute values are m-convex, AIP Conf. Proc., 1309(1) (2010), 861-863.
[21] H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.