A New Approach of Milne-type Inequalities Based on Proportional Caputo-Hybrid Operator
Abstract - 28


Convex functions
Fractional integrals
Milne-type inequalities
Proportional caputo-hybrid operator

How to Cite

Demir, İzzettin. (2023). A New Approach of Milne-type Inequalities Based on Proportional Caputo-Hybrid Operator. Journal of Advances in Applied & Computational Mathematics, 10, 102–119. https://doi.org/10.15377/2409-5761.2023.10.10


In this study, we first offer a novel integral identity using twice-differentiable convex mappings for the proportional Caputo-hybrid operator. Next, we demonstrate many integral inequalities related to the Milne-type integral inequalities for proportional Caputo-hybrid operator with the use of this newly discovered identity. Also, we present several examples along with their corresponding graphs in order to provide a better understanding of the newly obtained inequalities. Finally, we observe that the obtained results improve and generalize some of the previous results in the area of integral inequalities.

2010 Mathematics Subject Classification. 26D07, 26D10, 26A33



Shaikh AA, Hincal E, Ntouyas SK, Tariboon J, Tariq M. Some Hadamard-Type integral inequalities involving modified harmonic exponential type convexity. Axioms. 2023; 12(5): 454. https://doi.org/10.3390/axioms12050454

Tariq M, Ahmad H, Cesarano C, Abu-Zinadah H, Abouelregal AE, Askar S. Novel analysis of Hermiteâ€"Hadamard type integral inequalities via generalized exponential type m-convex functions. Mathematics. 2021; 10(1): 31. https://doi.org/10.3390/math10010031

Tariq M, Ahmad H, Sahoo SK, Nasir J, Awan SK. Some integral inequalities involving exponential type convex functions and applications. J Math Anal Model. 2021; 2(3): 62-76. https://doi.org/10.48185/jmam.v2i3.330

Sahoo SK, Tariq M, Ahmad H, Nasir J, Aydi H, Mukheimer A. New Ostrowski-type fractional integral inequalities via generalized exponential-type convex functions and applications. Symmetry 2021; 13(8): 1429. https://doi.org/10.3390/sym13081429

Tariq M. Hermite-Hadamard type inequalities via -harmonic exponential type convexity and appplications. Univers J Math Appl. 2021; 4(2): 59-69.

Tariq M, Nasir J, Sahoo SK, Mallah AA. A note on some Ostrowski type inequalities via generalized exponentially convexity. J Math Anal Model. 2021; 2(2): 1-15. https://doi.org/10.48185/jmam.v2i2.216

Dragomir SS, Agarwal RP, Cerone P. On Simpson's inequality and applications. J Inequal Appl. 2000; 5: 533-79. https://doi.org/10.1155/S102558340000031X

Alomari M, Darus M, Dragomir SS. New inequalities of Simpson's type for s-convex functions with applications. RGMIA Res Rep Coll. 2009; 12(4): Article 9.

Sar kaya MZ, Set E, Özdemir ME. On new inequalities of Simpson's type for functions whose second derivatives absolute values are convex. J Appl Math Stat Inform. 2013; 9(1): 37-45. https://doi.org/10.2478/jamsi-2013-0004

Budak H, Hezenci F, Kara H, Sar kaya MZ. Bounds for the error in approximating a fractional integral by Simpson's rule. Mathematics. 2023; 11(10): 1-16. https://doi.org/10.3390/math11102282

Hezenci F, Budak H, Kara H. New version of fractional Simpson type inequalities for twice differentiable functions. Adv Differ Equ 2021; 2021(460): 1-10. https://doi.org/10.1186/s13662-021-03615-2

Park J. On some integral inequalities for twice differentiable quasi-convex and convex functions via fractional integrals. Appl Math Sci. 2015; 9(62): 3057-69. https://doi.org/10.12988/ams.2015.53248

Sar kaya MZ, Set E, Özdemir ME. On new inequalities of Simpson's type for convex functions. RGMIA Res Rep Coll. 2010; 13(2): Article 2.

Alomari M, Liu Z. New error estimations for the Milne's quadrature formula in terms of at most first derivatives. Konuralp J Math. 2013; 1(1): 17-23.

Rom n-Flores H, Ayala V, Flores-Franuli A. Milne type inequality and interval orders. Comput Appl Math. 2021; 40(4): 1-15. https://doi.org/10.1007/s40314-021-01500-y

Budak H, Kösem P, Kara H. On new Milne-type inequalities for fractional integrals. J Inequal Appl. 2023; 2023(10): 1-15. https://doi.org/10.1186/s13660-023-02921-5

Ali MA, Zhang Z, Fe kan M. On some error bounds for Milne's formula in fractional calculus. Mathematics. 2023; 11(1): 1-11. https://doi.org/10.3390/math11010146

Bosch P, Rodriguez JM, Sigarreta JM. On new Milne-type inequalities and applications. J Inequal Appl. 2023; 2023(3): 1-18. https://doi.org/10.1186/s13660-022-02910-0

Budak H, Hyder A. Enhanced bounds for Riemann-Liouville fractional integrals: Novel variations of Milne inequalities. AIMS Mathematics. 2023; 8(12): 30760-76. https://doi.org/10.3934/math.20231572

Meftah B, Lakhdari A, Saleh W, K l çman A. Some new fractal Milne-type integral inequalities via generalized convexity with applications. Fractal Fract. 2023; 7(2): 1-15. https://doi.org/10.3390/fractalfract7020166

Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl. 2015; 1(2): 73-85.

Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci. 2016; 20(2): 763-9. https://doi.org/10.2298/TSCI160111018A

Sabzikar F, Meerschaert MM, Chen J. Tempered fractional calculus. J Comput Phys. 2015; 293: 14-28. https://doi.org/10.1016/j.jcp.2014.04.024

Diethelm K. The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Berlin, Germany: Springer; 2010. https://doi.org/10.1007/978-3-642-14574-2

Anderson DR, Ulness DJ. Newly defined conformable derivatives. Adv Dyn Syst Appl. 2015; 10(2): 109-37.

Günerhan H, Dutta H, Dokuyucu MA, Adel W. Analysis of a fractional HIV model with Caputo and constant proportional Caputo operators. Chaos Solitons Fractal. 2020;139: 1-19. https://doi.org/10.1016/j.chaos.2020.110053

Hajaj R, Odibat Z. Numerical solutions of fractional epidemic models with generalized Caputo-type derivatives. Phys Scr. 2023; 98(4): 045206. https://doi.org/10.1088/1402-4896/acbfef

Rahman G, Nisar KS, Abdeljawad T. Certain Hadamard proportional fractional integral inequalities. Mathematics. 2020; 8(4): 1-14. https://doi.org/10.3390/math8040504

Samko S, Kilbas A, Marichev O. Fractional integrals and derivatives: theory and applications. Switzerland; Philadelphia, Pa., USA: Gordon and Breach Science Publishers; 1993.

Baleanu D, Fernandez A, Akgül A. On a fractional operator combining proportional and classical differintegrals. Mathematics. 2020; 8(3): 360. https://doi.org/10.3390/math8030360

Sarikaya MZ. On Hermite-Hadamard type Inequalities for Proportional Caputo-Hybrid Operator. Konuralp J Math. 2023; 11(1): 31-9.

Sarikaya MZ. On Simpson type inequalities for proportional Caputo-Hybrid Operator. [Preprint · April 2023] Available from https://www.researchgate.net/publication/369950735

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Copyright (c) 2023 İzzettin Demir