Fractional integral Hermite-Hadamard inequalities type for MT-convex stochastic process
Oualid Rholamoualid.rholam@uit.ac.maDepartment of Logistics and MathematicsNational School of Applied SciencesUniversity Ibn TofailKenitra, MoroccoView full profile → , *Barmaki MohammedCorresponding authormohammed.barmaki@uit.ac.maDepartment of Analysis Modelling and SimulationUniversity Ibn TofailKenitra, MoroccoView full profile → , Driss Gretetedriss.gretete@uit.ac.maDepartment of Logisitics and MathematicsScience Faculty Ben M’sikUniversity Hassan IICasablanca, MoroccoView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 09 Nov 2022
- Published Online:
- 11 Apr 2025
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-1842
- Pages:
- 685–699
Abstract
Keywords
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References
[1] I. B. Lackovic and D. S. Mitrinivic, “Hermite and convexity,” Aequationes Math., vol. 28, pp. 229–232 (1985).
[2] P. Kumar, “Hermite-Hadamard inequalities and their applications in estimating moments,” in Inequality Theory and Applications, vol. 2. New York, NY, USA: Nova Science (2002). [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0898122104840136.
[3] E. A. Youness, “E-convex sets, E-convex functions, and Econvex programming,” J. Optim. Theory Appl., vol. 2, pp. 439–450 (1999), doi: 10.1023/A:1021792726715.
[4] J. El-Achky, D. Gretete, and M. Barmaki, “Inequalities of Hermite-Hadamard type for stochastic processes whose fourth derivatives absolute are quasi-convex, p-convex, s-convex and h-convex,” J. Interdiscip. Math., vol. 25, no. 4, pp. 987–1003 (2022). [Online]. Available: https://www.tandfonline.com/doi/abs/10.1080/09720502.2021.1887607.
[5] N. Uygun, E. Set, and A. Akdemir, “On New Simpson type Inequalities for generalized quasi-convex mappings,” in Xth International Statistics Days Conference, Giresun, Turkey, pp. 571–581 (2016).
[6] 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., vol. 23, pp. 1071–1076 (2010), doi: 10.1016/j.aml.2010.04.038.
[7] B. Nagy, “On a generalization of the Cauchy equation,” Aequationes Math., vol. 10, pp. 165–171 (1974), doi: 10.1007/BF01832853.
[8] K. Nikodem, “On convex stochastic processes,” Aequationes Math., vol. 20, pp. 184–197 (1980), doi: 10.1007/BF02190513.
[9] A. Skowronski, “On Wright-convex stochastic processes,” Ann. Math. Sil., vol. 9, pp. 29–32 (1995). [Online]. Available: http://www.sbc.org.pl/Content/34144/PDF/1995_04.pdf
[10] E. Set, M. Tomar, and S. Maden, “Hermite Hadamard type inequalities for s-convex stochastic processes in the second sense,” Turkish J. Anal. Number Theory, vol. 2, pp. 202–207 (2014). [Online]. Available: https://oaji.net/articles/2015/2533-1446015319.pdf.
[11] D. Kotrys, “Hermite-Hadamard inequality for convex stochastic processes,” Aequationes Math., vol. 83, pp. 143–151 (2012), doi: 10.1007/s00010-011-0090-1.
[12] J. J. Shynk, Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications. Hoboken, NJ, USA: Wiley (2013).
[13] A. Bain and D. Crisan, Fundamentals of Stochastic Filtering. New York, NY, USA: Springer-Verlag (2009).
[14] P. Devolder, J. Janssen, and R. Manemu, Basic Stochastic Processes: Mathematics and Statistics Series. London, UK: ISTE, John Wiley and Sons, pp. 380–444 (2015).
[15] T. Mikosch, Elementary Stochastic Calculus with Finance in View. Singapore: World Scientific Publishing (2010). [Online]. Available: https://doi.org/10.1142/3856.
[16] M. Shaked and J. Shantikumar, Stochastic Convexity and Its Applications. Tucson, AZ, USA: Univ. Arizona (1985).
[17] D. Kotrys, “Hermite-Hadamard inequality for convex stochastic processes,” Aequationes Math., vol. 83, pp. 143–151 (2012), doi: 10.1007/s00010-011-0090-1.
[18] R. Gorenflo and F. Mainardi, “Fractional calculus: Integral and differential equations of fractional order,” in CISM Courses and Lectures, vol. 378. Vienna, Austria: Springer, pp. 223–276 (1997). [Online]. Available: https://arxiv.org/abs/0805.3823.
[19] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier (2006). [Online]. Available: https://doi.org/10.1007/978-3-7091-2664-6_5.
[20] O. Rholam, M. Barmaki, and D. Gretete, ``Fractional integral inequalities of Hermite-Hadamard type for P-convex and quasi-convex stochastic process,’’ The Australian Journal of Mathematical Analysis and Applications, vol. 20, no. 10 (2023). [Online]. Available: https://ajmaa.org/cgi-bin/paper.pl?string=v20n1/V2011P10.tex.
[21] O. Rholam, M. Barmaki, and D. Gretete, “Hermite-Hadamard inequalities type using fractional integrals for \(\mathcal{MT}\)-convex stochastic process,” Malaysian Journal of Mathematical Sciences, vol. 17, pp. 473–485 (2023). [Online]. Available: https://mjms.upm.edu.my/lihatmakalah.php?kod=2023/September/17/3/473-485.
[22] S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. USA: John Wiley & Sons (1993).
[23] J. Wang and K. Qiu, “A fractional integral identity and its application to fractional Hermite-Hadamard type inequalities,” Journal of Interdisciplinary Mathematics, vol. 21, pp. 1–16 (2018). doi: 10.1080/09720502.2017.1400795.
[24] J. Wang and Z. Lin, “New Riemann-Liouville fractional Hermite-Hadamard inequalities via two kinds of convex functions,” Journal of Interdisciplinary Mathematics, vol. 20, pp. 357–382 (2010). doi: 10.1080/09720502.2014.914281.
[25] Y. Feng, “Refining Hermite-Hadamard integral inequality by two parameters,” Journal of Interdisciplinary Mathematics, vol. 21, pp. 743–746 (2018). doi: 10.1080/09720502.2018.1424093.
[26] U. N. Katugampola, “New approach to a generalized fractional integral,” Applied Mathematics and Computation, vol. 218, pp. 860–865 (2011). [Online]. Available: https://arxiv.org/abs/1010.0742.
[27] S. Mubeen and G. M. Habibullah, “k-Fractional integrals and application,” International Journal of Contemporary Mathematical Sciences, vol. 7, pp. 89–94 (2012). [Online]. Available: http://m-hikari.com/ijcms/ijcms-2012/1-4-2012/mubeenIJCMS1-4-2012-1.pdf.
[28] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70 (2014). doi: 10.1016/j.cam.2014.01.002.
[29] M. Kirane and B. T. Torebek, ``Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via fractional integrals,’’ Journal of Computational and Applied Mathematics, pp. 120-129 (2019). [Online]. Available: \url{https://doi.org/10.1016/j.cam.2018.12.030}.
[30] M. Z. Sarikaya and H. Yildirim, ``On generalization of the Riesz potential,’’ Indian Journal of Mathematics and Mathematical Sciences, vol. 3, pp. 231-235 (2007).
[31] R. Hussain, A. Ali, G. Gulshan, A. Latif, and M. Muddassar, ``Generalized co-ordinated integral inequalities for convex functions by way of k-fractional derivatives,’’Miskolc Mathematical Notes, Publications of the University of Miskolc.
[32] S. S. Dragomir and R. P. Agarwal, ``Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,’’ Appl. Math. Lett., vol. 11, pp. 91-95 (1998). [Online]. Available: https://doi.org/10.1016/S0893-9659(98)00086-X.
[33] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak, ``Hermite-Hadamard inequalities for fractional integrals and related fractional inequalities,’’ Mathematical and Computer Modelling, vol. 57, pp. 2403--2407 (2013). [Online]. Available: https://doi.org/10.1016/j.mcm.2011.12.048.




