Eastern Anatolian Journal of Science

2149-6137

Ağrı İbrahim Çeçen University

Operator Algebras and Functional Analysis

Operatör Cebirleri ve Fonksiyonel Analiz

Some Novel Fractional Integral Inequalities for m-Convex and (α,m)-Convex Functions

https://orcid.org/0000-0001-5970-1926

Aslan

Sinan

AGRI IBRAHIM CECEN UNIVERSITY

https://orcid.org/0000-0003-2466-0508

Akdemir

Ahmet Ocak

AGRI IBRAHIM CECEN UNIVERSITY

12 29 2025

11 1-2 1 8 06 17 2025 12 17 2025

2015

Eastern Anatolian Journal of Science

In this study, several novel integral inequalities are established for various classes of convex functions by employing the Caputo–Fabrizio fractional integral operator. Specifically, new inequalities are derived for m-convex and (α,m)-convex functions. The presented results generalize and extend existing inequalities in the literature, reducing to known outcomes for certain specific parameter values. The derivations rely on the fundamental properties of the Caputo–Fabrizio fractional operator, formal definitions of different types of convexity, and standard analytical techniques.

Caputo-Fabrizio fractional integral operator m-convex functions (Alpha m)- convex functions

ABDELJAWAD, THABET. On conformable fractional calculus. Journal of computational and Applied Mathematics, 2015, 279: 57-66.

ABDELJAWAD, THABET; BALEANU, DUMITRU. Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. J. Nonlinear Sci. Appl., 10 (2017), 1098–1107.

ABDELJAWAD, THABET; BALEANU, DUMITRU. On fractional derivatives with exponential kernel and their discrete versions. Reports on Mathematical Physics, 2017, 80.1: 11-27.

AKDEMİR, A. O.; ASLAN, S.; ÇAKALOĞLU, M.N. EKİNCİ, A. Some New Results for Different Kinds of Convex Functions CaputoFabrizio Fractional Operators. 4th International Conference on Mathematical and Related Sciences. Page (92) ICMRS 2021.

AKDEMİR, A. O.; ASLAN, S.; EKİNCİ, A. Novel Approaches for s-Convex Functions via Caputo-Fabrizio Fractional Integrals. Proceedings of IAM, 2022, 11.1: 3-16.

AKDEMIR, AHMET OCAK; BUTT, S.I; NADEEM, M; RAGUSA, M.A. New general variants of Chebyshev type inequalities via generalized fractional integral operators. Mathematics, 2021, 9.2: 122.

AKDEMIR, AHMET OCAK; EKINCI, ALPER; SET, ERHAN. Conformable fractional integrals and related new integral inequalities. 2017.

ATANGANA, ABDON; BALEANU, DUMITRU. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp. 763-769.

BUTT, SAAD IHSAN, UMAR, MUHAMMAD, RASHID, SAIMA, AKDEMIR, AHMET OCAK, CHU, YU-MING. New Hermite–Jensen–Mercer-type inequalities via k-fractional integrals. Advances in Difference Equations, 2020, 2020: 1-24. CAPUTO, MICHELE; FABRIZIO, MAURO. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation & Applications, 2015, 1.2: 73-85.

EKINCI, ALPER; OZDEMIR, MUHAMET. Some new integral inequalities via Riemann-Liouville integral operators. Applied and Computational Mathematics, 2019, 18.3.

GÜRBÜZ, MUSTAFA, AKDEMİR, AHMET OCAK, RASHID, SAIMA, SET, ERHAN. Hermite–Hadamard inequality for fractional integrals of Caputo–Fabrizio type and related inequalities. Journal of Inequalities and Applications, 2020, 2020: 1-10.

MIHESAN, V. G. (1993, May). A generalization of the convexity. In Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca, Romania.

ÖZDEMIR, M. E.; AVCI, M.; KAVURMACI, H. (2011). Hermite–Hadamard-type inequalities via (α,m)-convexity. Computers and Mathematics with Applications, 61(9), 2614-2620.

PECARIC, J.E.; TONG, Y.L. Convex functions, partial orderings, and statistical applications. Academic Press, 1992. RASHID, SAIMA, HAMMOUCH, ZAKIA, KALSOOM, HUMAIRA, ASHRAF, REHANA, CHU, YU MING. New investigation on the generalized K-fractional integral operators. Frontiers in Physics, 2020, 8: 25.

SAMKO, STEFAN G., et al. Fractional integrals and derivatives. Yverdon-les-Bains, Switzerland: Gordon and breach science publishers, Yverdon, 1993.

SET, ERHAN. New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals. Computers & Mathematics with Applications, 2012, 63.7: 1147-1154.

SET, ERHAN; AKDEMIR, AHMET OCAK; ÖZDEMIR, EMİN. M. Simpson type integral inequalities for convex functions via Riemann-Liouville integrals. Filomat, 2017, 31.14: 4415-4420.

TARIQ, MUHAMMAD, AHMAD, HIJAZ, SHAIKH, ABDUL GHAFOOR, SAHOO, SOUBHAGYA KUMAR, KHEDHER, KHALED MOHAMED, NGUYEN, TUAN GIA. New fractional integral inequalities for preinvex functions involving Caputo-Fabrizio operator. AIMS Mathematics, 2022, 7(3): 3440-3455. doi: 10.3934/math.2022191

TOADER, G. (1984, October). Some generalizations of the convexity. In Proceedings of the Colloquium on Approximation and Optimization (Vol. 329, p. 338). Cluj-Napoca, Romania: University of Cluj-Napoca.