Concerning the generalized Hermite-Hadamard integral inequality

Abstract

In the work we obtain some Hermite-Hadamard type inequalities for generalized fractional integrals for convex functions by employing a fractional integral operator, establishing, firstly, a basic identity that is used throughout the work. In addition, some classical integral inequalities are special cases of our main findings.

Keywords

• Hermite-Hadamard Inequality
• Generalized Fractional Integral
• Fractional Integral

References

1. REFERENCES
2. [1] Agarwal P, Jleli M, Tomar M. Certain Hermite-Hadamard type inequalities via generalized k-frac- tional integrals. J Inequal Appl 2017;1:1−10. [CrossRef]
3. [2] Ahmad B, Alsaedi A, Kirane M, Toberek BT. Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals. J Comput Appl Math 2019;353:120−129. [CrossRef]
4. [3] Ali MA, Nápoles JE, Kashuri A, Zhang Z. Fractional non conformable Hermite-Hadamard inequali-ties for generalized φ-convex functions. Fasc Math 2020;64:5−16.
5. [4] Bayraktar B, Butt SI, Shaokat S, Nápoles Valdés JE. New Hadamard-type inequalities via (s,m1,m2)-con-vex functions. Vestn Udmurt Univ Mat Mekhanika Komp'yuternye Nauki 2021;31:597−612. [CrossRef]
6. [5] Bayraktar B, Nápoles VJE, Rabossi F. On generaliza-tions of integral inequalities. Probl Anal Issues Anal 2022;11:3−23. [CrossRef]
7. [6] Bermudo S, K'orus P, Nápoles JE. On q-Hermite-Ha-damard inequalities for general convex functions. Acta Math Hung 2020;162:364−374. [CrossRef]
8. [7] Bohner M, Kashuri A, Mohammed PO, Nápoles VJE. Hermite-Hadamard-type inequalities for inte-grals arising in conformable fractional calculus. Hacet J Math Stat 2022;51:775−786.

9. [8] Bracamonte M, Gimenez J, Vivas M. Hermite-Hadamard-Fej'er type inequalities for (s,m)-Convex functions with modulus c, in second sense. Appl Math Inf Sci 2016;10:2045−2053. [CrossRef]
10. [9] Díaz R, Pariguan E. On hypergeometric func-tions and Pochhammer k-symbol. Divulg Mat 2007;15:179−192.
11. [10] Galeano Delgado JD, Lloreda J, Nápoles VJE, Pérez Reyes E. Certain integral inequalities of hermite-ha-damard type for h-convex functions. J Math Control Sci Appl 2021;7:129−140.
12. [11] Galeano J. D., Nápoles, J. E., and Pérez, E. (n/d), On a general formulation of the fractional oper-ator Riemann-Liouville and related inequalities, submitted.
13. [12] Hadamard J. Étude sur les propriétés des fonctions entiéres et en particulier d'une fonction considérée par Riemann. J Math Pures App 1893;9:171−216.
14. [13] Kashuri A, Nápoles V, Juan E, Ali MA, Din GMU. New Integral inequalities using quasi-convex func-tions via generalized ıntegral operators and their applications. Appl Math E-Notes 2022;22:221−231.
15. [14] Katugampola UN. New approach general-ized fractional integral. Appl Math Comput 2011;218:860−865. [CrossRef]
16. [15] Kórus P, Nápoles Valdés JE. On some integral inequalities for (h, m)-convex functions in a gener-alized framework. Carpathian J Math 2022. Preprint
17. [16] Mehmood S, Nápoles Valdés JE, Fatima N, Aslam W. Some integral inequalities via fractional derivatives. Adv Stud Euro Tbil Math J 2022;15:31−44. [CrossRef]
18. [17] Mohammed PO. Hermite-Hadamard inequalities for Riemann-Liouville fractional integrals of a con-vex function with respect to a monotone function. Math Meth Appl Sci 2019;44:2314−2324. [CrossRef]
19. [18] Mubeen S, Habibullah GM. K-fractional integrals and applications. Int J Contemp Math Sci 2012;7:89−94.
20. [19] Nápoles JE. Hermite-Hadamard inequality in generalized context, VI Colloquium on Applied Mathematics and II International Meeting of Applied Mathematics, Unimilitar, Bogotá, Colombia, November 11−13, 2020.
21. [20] Nápoles JE. New generalized fractional integral inequalities of Hermite-Hadamard type for harmon-ically convex functions, XVI International Meeting of Mathematics, Barranquilla, Colombia, November 17−20, 2020. [CrossRef]
22. [21] Nápoles Valdés JE. On the Hermite-Hadamard type inequalities involving generalized integrals. Contrib Math 2022;5:45−51. [CrossRef]
23. [22] Nápoles Valdés JE, Bayraktar B. On the generalized inequalities of the Hermite-Hadamard type. Filomat 2021;35:4917−4924. [CrossRef]
24. [23] Nápoles Valdés JE, Bayraktar B, Butt SI. New integral inequalities of Hermite-Hadamard type in a general-ized context. Punjab Univ J Math 2021;53:765−777.[CrossRef]
25. [24] Nápoles Valdés JE, Rabossi F, Ahmad H. Inequalities of the Hermite-Hadamard type, for functıons (h,m)-convex modified of the second typE. Commun Combin Cryptogr Computer Sci 2021;1:33−43.
26. [25] Nápoles Valdes JE, Rabossi F, Samaniego AD. Convex functions: Ariadne's thread or Charlotte's Spiderweb?, Adv Math Models Appl 2020;5:176−191.
27. [26] Nápoles Valdes JE, Rodríguez JM, Sigarreta JM. New Hermite-Hadamard type inequalities involv-ing non-conformable integral operators. Symmetry 2019;11:1108. [CrossRef] [27] Qi F, Guo BN. Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function. Rev Real Acad Cienc Exactas F'ıs Nat Ser A Mat 2017;111:425−434.[CrossRef]
28. [28] Qi F, Habib S, Mubeen S, Naeem MN. (2019), Generalized k-fractional conformable integrals and related inequalities. AIMS Mathematics 2019;4:343−358. [CrossRef]
29. [29] Rainville ED. Special Functions. New York: Macmillan Company; 1960.
30. [30] Ross, B. Fractional Calculus and its Applications. Berlin: Springer; 1975. [CrossRef]
31. [31] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications. 1st ed. Amsterdam: Taylor and Francis; 1993.
32. [32] Sarikaya MZ, Set E, Yaldiz H, Basak N. Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities. Math Comput Model 2013;57:2403−2407. [CrossRef]
33. [33] Vivas M. Jensen type inequalities for (s,m)-con-vex functions in second sense. Appl Math Inf Sci 2016;10:1689−1696. [CrossRef]
34. [34] Vivas M, Hernández J, Merentes N. New Hermite-Hadamard and Jensen type inequalities for h-Con-vex functions on fractal sets. Rev Colomb de Mat 2016;50:145−164. [CrossRef]
35. [35] Vivas-Cortez M, Kermausuor S, Nápoles Valdés JE. Hermite-Hadamard type inequalities for coor-dinated quasi-convex functions via generalized fractional integrals. In: Debnath P, Srivastava HM, Kumam P, Hazarika B, editors. Fixed point theory and fractional calculus.1st ed. Singapore: Springer; 2022. pp. 275−296. [CrossRef]
36. [36] Vivas-Cortez M, Kórus P, Nápoles Valdés JE. Some generalized Hermite-Hadamard-Fejér inequality for convex functions. Adv Differ Equ 2021;1:1−11. [CrossRef]
37. [37] Vivas M, Medina J. Hermite-Hadamard type inequalities for harmonically convex functions on n- coordinates. Appl Math Inf Sci Lett 2018;6:1−6.. [CrossRef]
38. [38] Vivas-Cortez M, Nápoles Valdés JE, Guerrero JA. Some Hermite-Hadamard weighted integral inequalities for (h,m)-convex modified functions. Appl Math Inf Sci 2022;16:25−33. [CrossRef]
39. [39] Yang ZH, Tian JF. Monotonicity and inequalities for the gamma function. J Inequal Appl 2017;1:1−15.[CrossRef]
40. [40] Yang ZH, Tian JF. Monotonicity and sharp inequal-ities related to gamma function. J Math Inequal 2018;12:1−22. [CrossRef]