Document Type : Research Paper


1 Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61110, Pakistan.

2 Department of Mathematical Engineering, Polytechnic University of Tirana, Tirana 1001, Albania.

3 Department of Mathematics, ITER, SOA University, Bhubaneswar 751030, India.


In this study, we established the Hermite-Hadamard type, Simpson type, Ostrowski type and midpoint type integral inequalities for the s-convex functions in the second sense via Katugampola fractional integrals. By using Katugampola fractional integral operators, we obtained several new identities and presented new results for the s-convex function in the second sense. We made connections of our results with various results recognized in the literature. Finally, applications to special means are examined to verify the efficiency of the established results.


Main Subjects

1. W. Afzal, S.M. Eldin, W. Nazeer and A.M. Galal, Some integral inequalities for harmonical cr-h-Godunova-Levin stochastic processes, AIMS Math., 8 (2023), pp. 13473-13491.
2. W. Afzal, M. Abbas, S.M. Eldin and Z.A. Khan, Some well-known inequalities for (h1, h2)-convex stochastic process via interval set inclusion relation, AIMS Math., 8 (2023), pp. 19913-19932.
3. G. Alberti, L. Ambrosio and P. Cannarsa, On the singularities of convex functions, Manuscripta Math., 76 (1992), pp. 421-435.
4. O. Almutairi and A. Kılıçman, New generalized Hermite-Hadamard inequality and related integral inequalities involving Katugampola type fractional integrals, Symmetry, 12 (2020).
5. M. Alomari, M. Darus, S.S. Dragomir and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23 (2010), pp. 1071-1076.
6. M. Alomari, M. Darus and S.S. Dragomir, New inequalities of Simpson’s Type for s-convex functions with applications, RGMIA Res. Rep. Coll., 12 (2009).
7. M. Alomari, M. Darus, S.S.Dragomir and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23 (2010), pp. 1071-1076.
8. M. Alomari, M. Darus and U.S. Kirmaci, Some inequalities of Hermite-Hadamard type for s-convex functions, Acta Math. Sci., 31 (2011), pp. 1643-1652.
9. M. Avci, H. Kavurmaci and M.E. Ozdemir, New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications, Appl. Math. Comput., 217 (2011), pp. 5171-5176.
10. S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10 (2009), pp. 1-12.
11. P.S. Bullen, D.S. Mitrinovic and P.M. Vasić, Means and their Inequalities, D. Reidel Publishing Co., Dordrecht, Netherlands, 1988.
12. Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9 (2010), pp. 493-497.
13. Z. Dahmani, L. Tabharit and S. Taf, New generalisations of Grüss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2 (2010), pp. 93-99.
14. M.R. Delavar and S.S. Dragomir, Hermite-Hadamard’s mid-point type inequalities for generalized fractional integrals, RACSAM REV R ACAD A, 114 (2020).
15. S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
16. S.S. Dragomir and S. Fitzpatrick, The Hadamard’s inequality for sconvex functions in the second sense, Demonstr. Math., 32 (1999), pp. 687-696.
17. S.S. Dragomir and T.M. Rassias, Ostrowski type inequalities and applications in numerical integration, Kluwer Academic Publishers:
Dordrecht, Boston, London, 2002.
18. A. Gozpinar, E. Set and S.S. Dragomir, Some generalized Hermite- Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex, Acta Math. Univ. Comen., 88 (2019), pp. 87-100.
19. H. Hudzik and L. Maligranda. Some remarks on s-convex functions, Aequationes Math., 48 (1994), pp. 100-111.
20. İ. İşcan, Generalization of different type integral inequalities for s-convex functions via fractional integrals, Appl. Anal., 93 (2013), pp. 1846-1862.
21. İ. İşcan, New estimates on generalization of some integral inequalities for s-convex functions and their applications, Int. J. Pure Appl. Math., 86 (2013), pp. 727-746.
22. A. Kashuri, New Integral Inequalities Pertaining Convex Functions And Their Applications, Matematichki Bilten. Bulletin Mathematique de la Societe des Mathematiciens de la Republique de Macedoine, 45 (2021).
23. A. Kashuri, P.O. Mohammed, T. Abdeljawad, F. Hamasalh and Y. Chu, New Simpson type integral inequalities for s-convex functions and their applications, Math. Probl. Eng., 2020 (2020), pp. 1-12.
24. U. Katugampola, New approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), pp. 1-15.
25. A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, Elsevier: Amsterdam. 2006.
26. N. Mehreen and M. Anwar, Integral inequalities for some convex functions via generalized fractional integrals, J. Inequal. Appl., 1 (2018), pp. 1-14.
27. C.P. Niculescu and L.E. Persson. Convex Functions and Their Applications, A Contemporary Approach, 2nd ed.; CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC; Springer: Cham, 2018.
28. J.E. Pečarić, F. Proschan and Y.L. Tong, Convex functions, partial orderings and statistical applications, Mathematics in Science and Engineering, 187; Academic Press, Inc.: Boston, MA, 1992.
29. A. Prudnikov, Y. Brychkov and O. Marichev, Integral and series In Elementary Functions, Nauka: Moscow. 1981.
30. R.U. Rahman, A.F. Al-Maaitah, M. Qousini, E.A. Az-Zo’bi, S.M. Eldin and M. Abuzar, New soliton solutions and modulation instability analysis of fractional Huxley equation, Results Phys., 44 (2023), pp. 106-163.
31. E. Rainville, Special Functions, The Mcmillan Company: New York. 1960.
32. M.Z. Sarıkaya, E. Set, H. Yaldız and N. Basak, Hermite- Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comp. Model, 57 (2013), pp. 2403-2407.
33. M.Z. Sarıkaya, E. Set and M.E. Ozdemir, On new inequalities of Simpson’s type for convex functions, RGMIA Res. Rep. Coll., 13 (2010), pp. 1-8.
34. M.Z. Sarikaya, E. Set and M.E. Ozdemir, On new inequalities of Simpson’s type for s-convex functions, Comput. Math. with Appl., 60 (2010), pp. 2191-2199.
35. E. Set, M. Noor, M. Awan and A. Gozpynar, Generalized Hermite- Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., 169 (2017), pp. 1-10.
36. E. Set, M.Z. Sarikaya, M.E. Ozdemir and H. Yildirim, The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results, J. Appl. Math. Stat. Inform., 10 (2014), pp. 69-83.
37. S. Sun, M.T. Khan, K. Al-Khaled, A. Raza, S.S. Abdullaev, S.U. Khan, N. Tamam and S.M. Eldin, Prabhakar fractional approach for enhancement of heat transfer due to hybrid nanomaterial with sinusoidal heat conditions, Case Stud. Therm. Eng., 49 (2023).
38. H. Yaldiz and A.O. Akdemir, Katugampola fractional integrals within the class of convex functions, Turk. J. Sci., 3 (2018), pp. 40-50.