Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals

Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals

We establish some new Generalized Hermite-Hadamard-type inequalities involving generalized fractional integrals for quasi-convex functions. Our results are suitable with the literature. The analysis used in the proofs is fairly elementary and based on the use of Hölder inequality and the power inequality.

___

  • Ali, A., Gulshan, G., Hussain, R., Latif, A. Muddassar, M. and Park, J., Generalized Inequalities of the type of Hermite-Hadamard-Fejer with Quasi-Convex Functions by way of k-Fractional Derivatives, J. Computational Analysis and applications, 22(7) (2017), 1208-1219.
  • Alomari, M., Darus, M., Dragomir, S.S., Inequalities of Hermite-Hadamard's type for functions whose derivatives absolute values are quasi-convex. RGMIA Res. Rep. Coll., 12: Supplement, Article 14 2009, 1-11.
  • Budak, H., Ertuğral, F., Sarıkaya, M.Z., New Generalization of Hermite-Hadamard Type Inequalities via Generalized Fractional Integrals. ResearchGate Article (2017). https://www.researchgate.net/publication/321760465.
  • Belarbi S., Dahmani Z., On some new fractional integral inequalities, J. Ineq. Pure Appl. Math., 10(3) (2009), Art. 86.
  • Carter, M., Brunt, B.V.: The Lebesgue-Stieljies Integral: A Practical Introduction. New York, Springer (2000).
  • Dahmani Z., New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9(4) (2010), 493-497.
  • Dragomir, S.S, Agarwal, R.P.: Two Inequalities for Di erentiable Mappings and Applications To Special Meansof Real Numbers and to Trapezoidal Formula. Appl. Math. Lett. 11 (1999), 91-95.
  • Dragomir, S.S., Pearce, C.E.M., Quasi-Convex Functions and Hadamard's Inequality. B. Aust. Math. Soc. 57 (1998), 377-385.
  • Dragomir, S.S., Pearce, C.E.M., Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA, Monographs. Victoria University, 2000.
  • Dragomir, S.S., Pecaric, J., and Persson, L.E., Some Inequalities of Hadamard Type. Soochow J. of Math. 21 (1995), 335-341.
  • Ertuğral, F. Sarıkaya, M.Z., Budak, H., On Hermite-Hadamard Type Inequalities Associated With The Generalized Fractional Integrals. ResearchGate Article (2019). https://www.researchgate.net/publication/334634529.
  • Gidergelmez, H.F., Akkurt, A., Yıldırım, H., Hermite-Hadamard Type Inequalities for Generalized Fractional Integrals via Strongly Convex Functions. KJM. 7 (2019), 268-273.
  • Gorenflo R., Mainardi F., Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer Verlag, Wien., (1997), 223-276.
  • Hadamard, J.: ėtude sur les propri ėtės des fonctions entiėres en particulier dűne fonction considėrėe par Riemann. Journal de mathėmatiques pures et appliquėes, 4e sėrie. 9 (1893), 171-216.
  • Hussain, R., Ali, A., Latif, A. and Gulshan, G., Some k-Fractional associates of Hermite-Hadamard's Inequality for Quasi-Convex Functions and Applications to Special Means, Fractional Differential Calculus, 7(2) (2017),301-309.
  • 10.. : Ion, D. A., Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Sci. Ser., 34 (2007), 82--87. Iscan I., New general integral inequalities for quasi-geometrically convex functions via fractional integrals, J. Inequal. Appl., (491) (2013), 1-15.
  • I scan, I.: Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals. Stud. Univ. Babes -Bolyai Math. 60 (2015), 355-366.
  • Khan, M.A., Khurshid, Y., Ali, T.: Hermite{Hadamard Inequality for Fractional Integrals via -Convex Functions. Acta Math. Univ. Comen. 86 (2017), 153-164.
  • K lbas, A.A., Srivastava, H.M., Trujillo J.J.: Theory and Applications of Fractional Di erential Equations. Elsevier, Amsterdam (2006).
  • Kunt, M., Karap nar, D., Turhan, S., İscan, İ.: The Left Riemann-Liouville Fractional Hermite-Hadamard Type Inequalities for Convex Functions. Math. Slovaca. 69 (2019), 773-784.
  • Miller S., Ross B., An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley. Soons. USA., 1993.
  • Mitrinovi c, D.S.: Analytic Inequalities. Springer, Berlin (1970).
  • Mitrinovi c, D.S., Pe cari c, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer, Dordrecht (1993).
  • Niculescu, C.P., Persson, L.E.: Convex Functions and Their Applications: A Contemporary Approach. Springer, New York (2006). Özdemir, M. E., Yıldız, Ç., Annals of the University of Craiova, Mathematics and Computer Science., 40 (2), (2013),167-173.
  • Pearce C. E. M., Quasi-convexity, fractional programming and extrenal traffic congestion, in "Frontiers in Global Optimization", Kluwer, Dordrecht, "Nonlinear Optimization and its Applications", 74 (2004), 403-409.
  • Pearce C. E. M. and Rubinov A. M., P- functions, quasi-convex functions and Hadamard type inequalities, J. Math. Anal. Applic., 240 (1999), 92-104.
  • Pe cari c, J.E., Proschan, F.,Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Boston, Academic Press (1992).
  • Ponstein, J., Seven types of convexity, SIAM Review 9 (1967), 115-119.
  • Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973).
  • Sar kaya, M.Z., Ertuğral, F.: On The Generalized Hermite-Hadamard Inequalities. Annals of University of Craiova, Math. Comp. Sci. Ser.,47 (1) (2020),193-213.
  • Sarıkaya M.Z., Ogunmez H., On new inequalities via Riemann-Liouville fractional integration, Abst. Appl. Anal., Art.ID 428983, (2012), 10 pages. http://dx.doi.org./10.1155/2012/428983.
  • Sar kaya, M.Z., Set, E., Yaldiz, H., Ba sak, N.: Hermite-Hadamard's Inequalities for Fractional Integrals and Related Fractional Inequalities. Math. Comput. Modell. 57 (2013), 2403-2407. Sarikaya, M.Z. and Yildirim, H., On generalization of the Riesz potential, Indian Jour. of Math. and Mathematical Sci. 3 (2007), no. 2, 231-235.
  • Set, E. and Çelik, B, Fractional Hermite-Hadamard Type Inequalities for Quasi-convex functions, Ordu Univ. J. Sci. Tech. 6, 1 (2016), 137--149.
  • Set, E., Karata s S.S., Khan, M.A.: Hermite{Hadamard Type Inequalities Obtained via Fractional Integrals for Di erentiable m-Convex and ( ;m)-Convex Functions. Int. J. Anal (2016). https://doi.org/10.1155/2016/4765691.
  • Set, E., Sar kaya, M.Z., Ozdemir, M.E., Yıldırım, H.: The Hermite-Hadamard's Inequality for Some Convex Functions via Fractional Integrals and Related Results. JAMSI. 10 (2014), 69-83.
  • Shi, D.-P., Xi, B.-Y., Qi, F.: Hermite-Hadamard Type Inequalities for Reimann-Liouville Fractional Integrals of (alpha;m)-Convex Functions. Fractional Di er. Calc. 4 (2014), 31-43.
  • Tun c, M.: On New Inequalities for h-Convex Functions via Riemann-Liouville Fractional Integration. Filomat, 27(2013), 559-565.
  • Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326 (2007), 303-311.
  • Yaldiz, H., Set, E.: Re nements Hermite-Hadamard-Fejer Type Inequalities For Generalized Fractional Integrals.ResearchGate Article (2018). https://www.researchgate.net/publication/323357856
  • Yıldırım, M.E., Sarıkaya, M.Z., Yldırım, H.: The Generalized Hermite-Hadamard-Fejer Type Inequalities For Generalized Fractional Integrals, ResearchGate Article (2018) https://www.researchgate.net/publication/322592667.
  • Zhao, D.M., Ali, A., Kashuri, A., Budak, H.: Generalized Fractional Integral Inequalities of Hermite-Hadamard Type for Harmonically Convex Functions. Adv. Di er. Equ. 1 (2020),1-14.