On Hermite-Hadamard Type Inequalities in Stochastic Fractional Calculus

Document Type : Original paper

Authors

1 Department of Mathematics, College of Dentistry, Diyala University, Diyala, Iraq

2 Department of Mathematics, College of Dentistry, University of Diyala, Diyala, Iraq

Abstract

This paper extends Hermite-Hadamard type inequalities within the framework of stochastic fractional calculus. We investigate how fractional integrals, which account for memory effects, interact with random processes. Our work presents three main contributions. First, we provide an error bound for approximating a standard integral of a smooth, deterministic function using stochastic fractional integrals. Second, we extend the well-known Hermite-Hadamard inequality, which applies to convex functions, to the setting of convex stochastic processes, showing how their expected values are bounded by these integrals. Finally, we derive specific mean-square error bounds when approximating a standard Brownian motion using its stochastic fractional integrals. These results enhance our understanding of stochastic fractional inequalities, offering new tools for analyzing complex systems influenced by both memory and randomness.

Keywords

Main Subjects

References

[1] M. Adil Khan, A. Iqbal, M. Suleman, and Y.M. Chu, Hermite–Hadamard type inequalities for fractional integrals via green’s function, J. Inequal. Appl. 2018 (2018), no. 1, Article number: 161. https://doi.org/10.1186/s13660-018-1751-6
[2] A. Babakhani, A generalized form of the Hermite-Hadamard-fejer type inequalities involving fractional integral for co-ordinated convex functions, Commun. Comb. Optim. 6 (2021), no. 1, 27–40.  https://doi.org/10.22049/cco.2020.26702.1132
[3] S.S. Dragomir and C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Science direct working paper (2003), no. S1574-0358, 04
[4] A. Kashuri, S.K. Sahoo, M. Aljuaid, M. Tariq, and M. De La Sen, Some new Hermite–Hadamard type inequalities pertaining to generalized multiplicative fractional integrals, Symmetry 15 (2023), no. 4, 868.  https://doi.org/10.3390/sym15040868
[5] Y.S. Mishura, Stochastic calculus for fractional brownian motion and related processes, Springer, 2008.
[6] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198, elsevier, 1998.
[7] M.Z. Sarikaya, New generalizations of Hermite-Hadamard type inequalities, Turk. J. Ineq 7 (2023), 29–39.
[8] N. Sharma, S.K. Mishra, and A. Hamdi, Hermite-Hadamard type inequality for ψ-Riemann-Liouville fractional integrals via preinvex functions, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 3333–3345. https://doi.org/10.22075/ijnaa.2021.21475.2262
[9] J.R. Wang, X. Li, and Y. Zhou, Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals via ψ-convex functions and applications to special means, Filomat 30 (2016), no. 5, 1143–1150.
[10] T. Zhou and T. Du, Certain fractional integral inclusions pertaining to interval-valued exponential trigonometric convex functions, J. Math. Inequal. 17 (2023), no. 1, 283–314. http://dx.doi.org/10.7153/jmi-2023-17-20 
Communications in Combinatorics and Optimization
Volume 11, Issue 3
September 2026
Pages 1059-1071

Files

Share

How to cite

Statistics