SOME NOVEL FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENT KINDS OF CONVEX FUNCTIONS

Year 2023, Volume: 9 Issue: 1, 27 - 32, 30.06.2023

Sinan Aslan

Abstract

In this paper, some novel integral inequalities for different kinds of
convex functions have been proved by using Caputo-Fabrizio fractional integral
operators. The findings includes several new integral inequalities h-convex
functions, s-convex functions in the second sense. We have used the properties
of Caputo-Fabrizio fractional operator, definition of different kinds of convex
functions and elemantery analysis methods.

Keywords

Caputo-Fabrizio fractional integral operator , h−convex functions , s−convex functions.

References

• 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. arXiv preprint arXiv:1607.00262, 2016.
• 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.; ÇAKALOĞLU, M.N. SET, E. New Hadamard Type Integral Inequalities via Caputo-Fabrizio Fractional Operators. 4th International Conference on Mathematical and Related Sciences. Page (91) 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, et al. 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. arXiv preprint arXiv:1602.03408, 2016.
• BRECKNER, WOLFGANG W. Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. Publ. Inst. Math.(Beograd)(NS), 1978, 23.37: 13-20.
• BUTT, SAAD IHSAN, et al. Hermite–Jensen–Mercer type inequalities for conformable integrals and related results. Advances in Difference Equations, 2020, 2020.1: 1-24.
• BUTT, SAAD IHSAN, et al. 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.
• 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, et al. Hermite–Hadamard inequality for fractional integrals of Caputo–Fabrizio type and related inequalities. Journal of Inequalities and Applications, 2020, 2020: 1-10.
• PEAJCARIAAC, JOSIP E.; TONG, YUNG LIANG. Convex functions, partial orderings, and statistical applications. Academic Press, 1992.
• RASHID, SAIMA, et al. New investigation on the generalized K-fractional integral operators. Frontiers in Physics, 2020, 8: 25.
• RASHID, SAIMA, et al. New multi-parametrized estimates having pth-order differentiability in fractional calculus for predominating ℏ-convex functions in Hilbert space. Symmetry, 2020, 12.2: 222.
• 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. 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, Emin M. Simpson type integral inequalities for convex functions via Riemann-Liouville integrals. Filomat, 2017, 31.14: 4415-4420.
• TARIQ, MUHAMMAD, et al. New fractional integral inequalities for preinvex functions involving caputo fabrizio operator. 2022.
• VAROŠANEC, SANJA. On h-convexity. Journal of Mathematical Analysis and Applications, 2007, 326.1: 303-311.DAVIS, S., MIRICK, D.K., STEVENS, R.G. (2002). Residential magnetic fields and the risk of breast cancer, Am J Epidemiol,155 (5), 446 – 454.