On generalizations of trapezoid and Bullen type inequalities based on generalized fractional integrals

Hüseyin Budak; Fatma Ertuğral; Muhammad Aamir Ali; Candan Can Bilişik; Mehmet Zeki Sarikaya; Kamsing Nonlaopon; Hüseyin Budak; Fatma Ertuğral; Muhammad Aamir Ali; Candan Can Bilişik; Mehmet Zeki Sarikaya; Kamsing Nonlaopon

doi:10.3934/math.2023094

AIMS Mathematics

2023, Volume 8, Issue 1: 1833-1847. doi: 10.3934/math.2023094

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Research article

On generalizations of trapezoid and Bullen type inequalities based on generalized fractional integrals

1.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 06 August 2022 Revised: 16 September 2022 Accepted: 28 September 2022 Published: 25 October 2022
MSC : 26D07, 26D10, 26D15

In this paper, we establish an integral identity involving differentiable functions and generalized fractional integrals. Then, using the newly established identity, we prove some new general versions of Bullen and trapezoidal type inequalities for differentiable convex functions. The main benefit of the newly established inequalities is that they can be converted into similar inequalities for classical integrals, Riemann-Liouville fractional integrals, $ k $-Riemann-Liouville fractional integrals, Hadamard fractional integrals, etc. Moreover, the inequalities presented in the paper are extensions of several existing inequalities in the literature.
- Hermite-Hadamard inequality,
- fractional integral operators,
- convex function
Citation: Hüseyin Budak, Fatma Ertuğral, Muhammad Aamir Ali, Candan Can Bilişik, Mehmet Zeki Sarikaya, Kamsing Nonlaopon. On generalizations of trapezoid and Bullen type inequalities based on generalized fractional integrals[J]. AIMS Mathematics, 2023, 8(1): 1833-1847. doi: 10.3934/math.2023094

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Abstract

In this paper, we establish an integral identity involving differentiable functions and generalized fractional integrals. Then, using the newly established identity, we prove some new general versions of Bullen and trapezoidal type inequalities for differentiable convex functions. The main benefit of the newly established inequalities is that they can be converted into similar inequalities for classical integrals, Riemann-Liouville fractional integrals, $ k $-Riemann-Liouville fractional integrals, Hadamard fractional integrals, etc. Moreover, the inequalities presented in the paper are extensions of several existing inequalities in the literature.

References

[1]	M. Alomari, M. Darus, S. S. Dragomir, New inequalities of Simpson's type for $\mathit{s}$-convex functions with applications, Res. Rep. Collect., 12 (2009).
[2]	M. U. Awan, S. Talib, Y. M. Chu, M. A. Noor, K. I. Noor, Some new refinements of Hermite-Hadamard-type inequalities involving-Riemann-Liouville fractional integrals and applications, Math. Probl. Eng., 2020 (2020), 3051920. https://doi.org/10.1155/2020/3051920 doi: 10.1155/2020/3051920
[3]	H. Budak, F. Hezenci, H. Kara, On parameterized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integral, Math. Method. Appl. Sci., 44 (2021), 12522–12536. https://doi.org/10.1002/mma.7558 doi: 10.1002/mma.7558
[4]	H. Budak, P. Agarwal, New generalized midpoint type inequalities for fractional integral, Miskolc Math. Notes, 20 (2019), 781–793. https://doi.org/10.18514/MMN.2019.2525 doi: 10.18514/MMN.2019.2525
[5]	H. Budak, R. Kapucu, New generalization of midpoint type inequalities for fractional integral, An. Ştiint. Univ Al. I. Cuza Iaşi. Mat. (N.S), 67 (2021). https://doi.org/10.47743/anstim.2021.00009 doi: 10.47743/anstim.2021.00009
[6]	P. S. Bullen, Error estimates for some elementary quadrature rules, Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 602/633 (1978), 97–103.
[7]	M. Çakmak, Some Bullen-type inequalities for conformable fractional integrals, Gen. Math., 28 (2020), 3–17.
[8]	S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X doi: 10.1016/S0893-9659(98)00086-X
[9]	T. Du, Y. Li, Z. Yang, A generalization of Simpson's inequality via differentiable mapping using extended $(s, m)$-convex functions, Appl. Math. Comput., 293 (2017), 358–369. https://doi.org/10.1016/j.amc.2016.08.045 doi: 10.1016/j.amc.2016.08.045
[10]	T. Du, C. Luo, Z. Cao, On the Bullen-type inequalities via generalized fractional integrals and their applications, Fractals, 29 (2021). https://doi.org/10.1142/S0218348X21501887 doi: 10.1142/S0218348X21501887
[11]	S. Erden, M. Z. Sarikaya, Generalized Bullen-type inequalities for local fractional integrals and its applications, Palestine J. Math., 9 (2020), 945–956.
[12]	F. Ertuğral, M. Z. Sarikaya, Simpson type integral inequalities for generalized fractional integral, RACSAM Rev. R. Acad. A, 113 (2019), 3115–3124. https://doi.org/10.1007/s13398-019-00680-x doi: 10.1007/s13398-019-00680-x
[13]	F. Ertuğral, M. Z. Sarikaya, H. Budak, On Hermite-Hadamard type inequalities associated with the generalized fractional integrals, Filomat, 2022, In press.
[14]	G. Farid, A. U. Rehman, M. Zahra, On Hadamard inequalities for $k$-fractional integrals, Nonlinear Funct. Anal. Appl., 21 (2016), 463–478.
[15]	R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Springer Verlag, Wien, 1997.
[16]	İ. İşcan, Hermite-Hadamard and Simpson-like type inequalities for differentiable harmonically convex functions, J. Math., 2014 (2014). https://doi.org/10.1155/2014/346305 doi: 10.1155/2014/346305
[17]	M. Iqbal, S. Qaisar, M. Muddassar, A short note on integral inequality of type Hermite-Hadamard through convexity, J. Comput. Anal. Appl., 21 (2016), 946–953.
[18]	A. Kashuri, R. Liko, Generalized trapezoidal type integral inequalities and their applications, J. Anal., 28 (2020), 1023–1043. https://doi.org/10.1007/s41478-020-00232-2 doi: 10.1007/s41478-020-00232-2
[19]	M. A. Khan, A. Iqbal, M. Suleman, Y. M. Chu, Hermite-Hadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl., 2018 (2018), 1–15. https://doi.org/10.1186/s13660-018-1751-6 doi: 10.1186/s13660-018-1751-6
[20]	M. A. Khan, T. Ali, S. S. Dragomir, M. Z. Sarikaya, Hermite-Hadamard type inequalities for conformable fractional integrals, RACSAM Rev. R. Acad. A, 112 (2018), 1033–1048. https://doi.org/10.1007/s13398-017-0408-5 doi: 10.1007/s13398-017-0408-5
[21]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[22]	U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147 (2004), 137–146. https://doi.org/10.1016/S0096-3003(02)00657-4 doi: 10.1016/S0096-3003(02)00657-4
[23]	M. A. Latif, Inequalities of Hermite-Hadamard type for functions whose derivatives in absolute value are convex with applications, Arab J. Math. Sci., 21 (2015), 84–97. https://doi.org/10.1016/j.ajmsc.2014.01.002 doi: 10.1016/j.ajmsc.2014.01.002
[24]	M. Matloka, Some inequalities of Simpson type for $h$-convex functions via fractional integrals, Abstr. Appl. Anal., 2015 (2015). https://doi.org/10.1155/2015/956850 doi: 10.1155/2015/956850
[25]	F. C. Mitroi, M. V. Mihai, Hermite-Hadamard type inequalities obtained via Riemann-Liouville fractional calculus, Acta Math. Univ. Comen., 83 (2014), 209–215.
[26]	S. Mubeen, G. M. Habibullah, $k$-fractional integrals and application, Int. J. Contemp. Math. Sci., 7 (2012), 89–94.
[27]	M. E. Ozdemir, A. O. Akdemir, H. Kavurmacı, On the Simpson's inequality for convex functions on the coordinates, Turk. J. Anal. Number Theor., 2 (2014), 165–169.
[28]	S. Qaisar, S. Hussain, On Hermite-Hadamard type inequalities for functions whose first derivative absolute values are convex and concave, Fasciculi Math., 58 (2017), 155–166. https://doi.org/10.1515/fascmath-2017-0011 doi: 10.1515/fascmath-2017-0011
[29]	J. Park, On Simpson-like type integral inequalities for differentiable preinvex functions, Appl. Math. Sci., 7 (2013), 6009–6021. https://doi.org/10.12988/ams.2013.39498 doi: 10.12988/ams.2013.39498
[30]	J. Park, On some integral inequalities for twice differentiable quasi-convex and convex functions via fractional integrals, Appl. Math. Sci., 9 (2015), 3057–3069. https://doi.org/10.12988/ams.2015.53248 doi: 10.12988/ams.2015.53248
[31]	J. E. Pečarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings and statistical applications, Academic Press, Boston, 1992.
[32]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite-Hadamard's inequalities for fractional integrals and relatedfractional 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
[33]	M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson's type for convex functions, RGMIA Res. Rep. Collect., 13 (2010). https://doi.org/10.1016/j.camwa.2010.07.033 doi: 10.1016/j.camwa.2010.07.033
[34]	M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson's type for $s$-convex functions, Comput. Math. Appl., 60 (2020), 2191–2199. https://doi.org/10.1016/j.camwa.2010.07.033 doi: 10.1016/j.camwa.2010.07.033
[35]	M. Z. Sarikaya, H. Budak, Some integral inequalities for local fractional integrals, Int. J. Anal. Appl., 14 (2017), 9–19. https://doi.org/10.1016/j.amc.2015.11.096 doi: 10.1016/j.amc.2015.11.096
[36]	M. Z. Sarikaya, F. Ertugral, On the generalized Hermite-Hadamard inequalities, Ann. Univ. Craiova-Mat., 47 (2020), 193–213. https://doi.org/10.52846/ami.v47i1.1139 doi: 10.52846/ami.v47i1.1139
[37]	E. Set, J. Choi, A. Gözpinar, Hermite-Hadamard type inequalities for the generalized k-fractional integral operators, J. Inequal. Appl., 2017 (2017), 1–17. https://doi.org/10.1186/s13660-017-1476-y doi: 10.1186/s13660-017-1476-y
[38]	M. Vivas-Cortez, M. A. Ali, A. Kashuri, H. Budak, Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions, AIMS Math., 6 (2021), 9397–9421. https://doi.org/10.3934/math.2021546 doi: 10.3934/math.2021546
[39]	D. Zhao, M. A. Ali, A. Kashuri, H. Budak, Generalized fractional integral inequalities of Hermite-Hadamard type for harmonically convex functions, Adv. Differ. Equ., 2020 (2020), 1–14. https://doi.org/10.1186/s13662-020-02589-x doi: 10.1186/s13662-020-02589-x
[40]	D. Zhao, M. A. Ali, A. Kashuri, H. Budak, M. Z. Sarikaya, Hermite-Hadamard-type inequalities for the interval-valued approximately $h$-convex functions via generalized fractional integrals, J. Inequal. Appl., 2020 (2020), 1–38. https://doi.org/10.1186/s13660-020-02488-5 doi: 10.1186/s13660-020-02488-5

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