Fractional Chebyshev-Type Inequalities for Synchronous Functions via Unified Fractional Remainder Operator
DOI:
https://doi.org/10.15377/2409-5761.2026.13.12Keywords:
Chebyshev inequality, Chebyshev functional, Synchronous functions, Weighted integral inequalities, Unified fractional remainder operatorAbstract
In this paper, we establish new fractional Chebyshev-type inequalities for synchronous functions by means of a unified fractional remainder operator. Motivated by the classical Chebyshev integral inequality and its extensions, we investigate how such inequalities can be formulated within a generalized fractional integral framework. To this end, we introduce a normalized form of the unified fractional remainder operator and define an associated fractional Chebyshev functional. Using the nonnegativity and normalization properties of the corresponding fractional kernel, we derive a double-integral representation of this functional. This representation provides the key tool for establishing a fractional analogue of the classical two-function Chebyshev integral inequality for synchronous functions. Furthermore, we extend the approach to obtain a multi-function Chebyshev-type inequality within the same fractional framework. The assumptions ensuring the integrability of the functions and their products are taken into account in establishing these results. Finally, we demonstrate that the classical Chebyshev integral inequality is recovered as a particular case through an appropriate choice of the parameters of the unified operator. These findings extend the applicability of Chebyshev-type inequalities to the proposed fractional setting and provide a framework for deriving related inequalities.
Mathematics Subject Classification: 26A33; 26D07; 26D10; 33B15.
References
Dehghan M, Manafian J, Saadatmandi A. Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ. 2010; 26(2): 448-79. https://doi.org/10.1002/num.20460
Dehghan M, Manafian J, Saadatmandi A. The solution of the linear fractional partial differential equations using the homotopy analysis method. Z Naturforsch A. 2010; 65(11): 935-44. https://doi.org/10.1515/zna-2010-1106
Manafian J, Lakestani M. A new analytical approach to solve some of the fractional-order partial differential equations. Indian J Phys. 2017; 91(3): 243-58. https://doi.org/10.1007/s12648-016-0912-z
Rezazadeh H, Khodadad FS, Manafian J. New structure for exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation via conformable fractional derivative. Appl Appl Math. 2017; 12(1): 26.
Lakestani M, Manafian J. Analytical treatment of nonlinear conformable time-fractional Boussinesq equations by three integration methods. Opt Quantum Electron. 2018; 50(1): 4. https://doi.org/10.1007/s11082-017-1268-0
Qawaqneh H, Manafian J, Alharthi M, Alrashedi Y. Stability analysis, modulation instability, and beta-time fractional exact soliton solutions to the van der Waals equation. Mathematics. 2024; 12(14): 2257. https://doi.org/10.3390/math12142257
Meena RK, Kumar S. Solution of fractional order SIR epidemic model using residual power series method. Palest J Math. 2022; 11(3): 12-24.
Meena RK, Kumar S. A study on fractional SIS epidemic model using RPS method. In: Proceedings of the International Conference on Mathematics and Computing; 2023 Jan 6. Singapore: Springer Nature Singapore; 2023. p. 293-309. https://doi.org/10.1007/978-981-99-3080-7_22
Meena RK, Kumar S. A study on fractional SIR epidemic model with vital dynamics and variable population size using the residual power series method. J Comput Anal Appl. 2024;33(1): 235-52.
Meena RK, Kumar S. A study on fractional mathematical model for cancer chemotherapy using the residual power series method. Comput Methods Biomech Biomed Engin. 2026; 1-9 (epub ahead). https://doi.org/10.1080/10255842.2026.2664772
Zaak MDA, Maazouz K, Rodríguez-López R, Sabit S. A mathematical insight of fractional logistic equation of variable order on finite intervals. Filomat. 2025; 39(26): 9179-89. https://doi.org/10.2298/FIL2526179Z
Bouharket B, Azzouz N, Budak H. Generalized refinement of Hermite-Hadamard inequality. J Inequal Appl. 2025; 2025: 132. https://doi.org/10.1186/s13660-025-03386-4
Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier; 2006. (North-Holland Mathematics Studies; vol. 204). https://doi.org/10.1016/S0304-0208(06)80001-0
Chebyshev PL. Sur les expressions approximatives des intégrales définies par les autres prises entre les mêmes limites. Proc Math Soc Kharkov. 1882; 2: 93-8.
Belarbi S, Dahmani Z. On some new fractional integral inequalities. Int J Math Anal. 2010; 4: 185-91.
Akdemir AO, Butt SI, Nadeem M, Ragusa MA. New general variants of Chebyshev type inequalities via generalized fractional integral operators. Mathematics. 2021; 9(2): 122. https://doi.org/10.3390/math9020122
Usta F, Budak H, Sarikaya MZ. Some new Chebyshev type inequalities utilizing generalized fractional integral operators. AIMS Math. 2020; 5(2): 1147-61. https://doi.org/10.3934/math.2020079
Guzmán PM, Kórus P, Nápoles Valdés JE. Generalized integral inequalities of Chebyshev type. Fractal Fract. 2020;4(2):10. https://doi.org/10.3390/fractalfract4020010
Set E, Özdemir ME, Demirbaş S. Chebyshev type inequalities involving extended generalized fractional integral operators. AIMS Math. 2020; 5(4): 3573-83. https://doi.org/10.3934/math.2020232
Srivastava HM, Kashuri A, Mohammed PO, Alsharif AM, Guirao JLG. New Chebyshev type inequalities via a general family of fractional integral operators with a modified Mittag-Leffler kernel. AIMS Math. 2021; 6(10): 11167-86. https://doi.org/10.3934/math.2021648
Kórus P, Nápoles Valdés JE. A review of the Chebyshev inequality pertaining to fractional integrals. Mathematics. 2025; 13(7): 1137. https://doi.org/10.3390/math13071137
Houas M, Dahmani Z, Sarikaya MZ. Some integral inequalities for (k,s) -Riemann-Liouville fractional operators. J Interdiscip Math. 2018; 21(7-8): 1575-85. https://doi.org/10.1080/09720502.2018.1478768
Iqbal SA, Mubeen SH, Tomar MU. On Hadamard k-fractional integrals. J Fract Calc Appl. 2018; 9(2): 255-67.
Nisar KS, Rahman G, Mehrez K. Chebyshev type inequalities via generalized fractional conformable integrals. J Inequal Appl. 2019; 2019: 245. https://doi.org/10.1186/s13660-019-2197-1
Benaissa B, Azzouz N, Sarikaya MZ. On some Tchebyshev-type inequalities involving the k-weighted fractional operator. Mathematica. 2025; 67(2): 181-96. https://doi.org/10.24193/mathcluj.2025.2.03
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