On Hermite-Hadamard type inequalities for co-ordinated convex function via conformable fractional integrals

Mehmet Eyüp Kiriş; Miguel Vivas-Cortez; Gözde Bayrak; Tuğba Çınar; Hüseyin Budak; Mehmet Eyüp Kiriş; Miguel Vivas-Cortez; Gözde Bayrak; Tuğba Çınar; Hüseyin Budak

doi:10.3934/math.2024502

AIMS Mathematics

2024, Volume 9, Issue 4: 10267-10288. doi: 10.3934/math.2024502

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On Hermite-Hadamard type inequalities for co-ordinated convex function via conformable fractional integrals

1.
Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyonkarahisar, Türkiye
2.
Pontificia Universidad Católica del Ecuador, Facultad de Ciencias Naturales y Exactas, Escuela de Ciencias Físicas y Matemáticas, Sede Quito, Ecuador
3.
Afyon Kocatepe University Graduate School of Natural and Applied Sciences, Afyonkarahisar, Türkiye
4.
Department of Mathematics, Faculty of Science and Arts, Dü zce University, Düzce, Türkiye

Received: 20 December 2023 Revised: 07 February 2024 Accepted: 09 February 2024 Published: 14 March 2024
MSC : 26A33, 26A51, 26B25, 26D10, 26D15

In this study, some new Hermite-Hadamard type inequalities for co-ordinated convex functions were obtained with the help of conformable fractional integrals. We have presented some remarks to give the relation between our results and earlier obtained results. Moreover, an identity for partial differentiable functions has been established. By using this equality and concept of co-ordinated convexity, we have proven a trapezoid type inequality for conformable fractional integrals.
- convex functions,
- co-ordinated convex mapping,
- Hermite-Hadamard inequality,
- conformable fractional integral,
- trapezoid type inequalities
Citation: Mehmet Eyüp Kiriş, Miguel Vivas-Cortez, Gözde Bayrak, Tuğba Çınar, Hüseyin Budak. On Hermite-Hadamard type inequalities for co-ordinated convex function via conformable fractional integrals[J]. AIMS Mathematics, 2024, 9(4): 10267-10288. doi: 10.3934/math.2024502

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Abstract

In this study, some new Hermite-Hadamard type inequalities for co-ordinated convex functions were obtained with the help of conformable fractional integrals. We have presented some remarks to give the relation between our results and earlier obtained results. Moreover, an identity for partial differentiable functions has been established. By using this equality and concept of co-ordinated convexity, we have proven a trapezoid type inequality for conformable fractional integrals.

References

[1]	S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000.
[2]	J. E. Pečarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings and statistical applications, Boston: Academic Press, 1992.
[3]	S. S. Dragomir, On the Hadamard's inequlality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., 5 (2001), 775–788. https://doi.org/10.11650/twjm/1500574995 doi: 10.11650/twjm/1500574995
[4]	A. Akkurt, M. Z. Sarikaya, H. Budak, H. Yildirim, On the Hadamard's type inequalities for co-ordinated convex functions via fractional integrals, J. King Saud Univ. Sci., 29 (2017), 380–387. https://doi.org/10.1016/j.jksus.2016.06.003 doi: 10.1016/j.jksus.2016.06.003
[5]	A. Akkurt, M. Z. Sarikaya, H. Budak, H. Yildirim, On the Hermite-Hadamard type inequalities for co-ordinated convex functions, Appl. Comput. Math., 20 (2021), 408–420.
[6]	M. K. Bakula, An improvement of the Hermite-Hadamard inequality for functions convex on the coordinates, Aust. J. Math. Anal. Appl., 11 (2014), 1–7.
[7]	M. Alomari, M. Darus, The Hadamards inequality for $ s $-convex function of 2-variables on the coordinates, Int. J. Math. Anal., 2 (2008), 629–638.
[8]	M. Vivas-Cortez, H. Kara, H. Budak, M. A. Ali, S. Chasreechai, Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions, Open Math., 20 (2022), 1887–1903. https://doi.org/10.1515/math-2022-0477 doi: 10.1515/math-2022-0477
[9]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[10]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[11]	I. Podlubny, Fractional differential equations, San Diego, CA: Academic Press, 1999.
[12]	R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[13]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
[14]	A. A. Abdelhakim, The flaw in the conformable calculus: It is conformable because it is not fractional, Fract. Calc. Appl. Anal., 22 (2019), 242–254. https://doi.org/10.1515/fca-2019-0016 doi: 10.1515/fca-2019-0016
[15]	T. S. Du, Y. Peng, Hermite-Hadamard type inequalities for multiplicative Riemann-Liouville fractional integrals, J. Comput. Appl. Math., 440 (2024), 115582. https://doi.org/10.1016/j.cam.2023.115582 doi: 10.1016/j.cam.2023.115582
[16]	A. A. Hyder, A. A. Almoneef, H. Budak, M. A. Barakat, On new fractional version of generalized Hermite-Hadamard inequalities, Mathematics, 10 (2022), 3337. https://doi.org/10.3390/math10183337 doi: 10.3390/math10183337
[17]	T. U. Khan, M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389. https://doi.org/10.1016/j.cam.2018.07.018 doi: 10.1016/j.cam.2018.07.018
[18]	M. Vivas-Cortez, S. Kermausuor, J. E. N. Valdés, Ostrowski type inequalities for conformable fractional calculus via a parameter, In: Advanced mathematical analysis and its applications, Chapman and Hall/CRC, 2023.
[19]	F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 247. https://doi.org/10.1186/s13662-017-1306-z doi: 10.1186/s13662-017-1306-z
[20]	M. Bozkurt, A. Akkurt, H. Yildirim, Conformable derivatives and integrals for the functions of two variables, Konuralp J. Math., 9 (2021), 49–59.
[21]	E. Set, J. Choi, A. Gözpinar, Hermite-Hadamard type inequalities involving nonlocal conformable fractional integrals, Malaysian J. Math. Sci., 15 (2021), 33–43.
[22]	M. A. Latif, M. W. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinates, Int. Math. Forum, 4 (2009), 2327–2338.
[23]	M. Z. Sarikaya, On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integr. Transf. Spec. Funct., 25 (2014), 134–147. https://doi.org/10.1080/10652469.2013.824436 doi: 10.1080/10652469.2013.824436
[24]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
[25]	M. Z. Sarikaya, A. Akkurt, H. Budak, M. E. Yildirim, H. Yildirim, Hermite-Hadamard's inequalities for conformable fractional integrals, Int. J. Optim. Control Theor. Appl., 9 (2019), 49–59. https://doi.org/10.11121/ijocta.01.2019.00559 doi: 10.11121/ijocta.01.2019.00559
[26]	M. W. Alomari, M. Darus, Fejér inequality for double integrals, Facta Univ. Math. Inform., 24 (2009), 15–28.
[27]	M. Z. Sarikaya, E. Set, M. E. Özdemir, S. S. Dragomir, New some Hadamard's type inequalities for co-ordinated convex functions, Tamsui Oxf. J. Inf. Math. Sci., 28 (2010), 137–152.

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