New Simpson type inequalities for twice differentiable functions via generalized fractional integrals

Xuexiao You; Fatih Hezenci; Hüseyin Budak; Hasan Kara; Xuexiao You; Fatih Hezenci; Hüseyin Budak; Hasan Kara

doi:10.3934/math.2022218

AIMS Mathematics

2022, Volume 7, Issue 3: 3959-3971. doi: 10.3934/math.2022218

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Research article

New Simpson type inequalities for twice differentiable functions via generalized fractional integrals

1.
School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
2.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

Received: 26 October 2021 Accepted: 06 December 2021 Published: 10 December 2021
MSC : 26A33, 26D07, 26D10, 26D15

Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type inequalities for functions whose second derivatives in absolute value are convex.
- Simpson type inequalities,
- generalized fractional integrals,
- convex functions
Citation: Xuexiao You, Fatih Hezenci, Hüseyin Budak, Hasan Kara. New Simpson type inequalities for twice differentiable functions via generalized fractional integrals[J]. AIMS Mathematics, 2022, 7(3): 3959-3971. doi: 10.3934/math.2022218

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Abstract

Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type inequalities for functions whose second derivatives in absolute value are convex.

References

[1]	M. Alomari, M. Darus, S. Dragomir, New inequalities of Simpson's type for $\mathit{s}$-convex functions with applications, Res. Rep. Coll., 12 (2009), 9.
[2]	M. Sarikaya, E. Set, M. Özdemir, On new inequalities of Simpson's type for $s$-convex functions Comput. Math. Appl., 60 (2010), 2191–2199. doi: 10.1016/j.camwa.2010.07.033. doi: 10.1016/j.camwa.2010.07.033
[3]	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. doi: 10.1016/j.amc.2016.08.045. doi: 10.1016/j.amc.2016.08.045
[4]	İ. İşcan, Hermite-Hadamard, Simpson-like type inequalities for differentiable harmonically convex functions, J. Math., 2014 (2014), 346305. doi: 10.1155/2014/346305. doi: 10.1155/2014/346305
[5]	M. Matloka, Some inequalities of Simpson type for h-convex functions via fractional integrals, Abstr. Appl. Anal., 2015 (2015), 956850. doi: 10.1155/2015/956850. doi: 10.1155/2015/956850
[6]	M. E. Ozdemir, A. O. Akdemir, H. Kavurmacı, On the Simpson's inequality for convex functions on the coordinates, Turkish Journal of Analysis and Number Theory, 2 (2014), 165–169. doi: 10.12691/tjant-2-5-2. doi: 10.12691/tjant-2-5-2
[7]	J. Park, On Simpson-like type integral inequalities for differentiable preinvex functions, Applied Mathematical Sciences, 7 (2013), 6009–6021. doi: 10.12988/ams.2013.39498. doi: 10.12988/ams.2013.39498
[8]	J. Chen, X. Huang, Some new inequalities of Simpson's type for $s$-convex functions via fractional integrals, Filomat, 31 (2017), 4989–4997. doi: 10.2298/FIL1715989C. doi: 10.2298/FIL1715989C
[9]	M. Iqbal, S. Qaisar, S. Hussain, On Simpson's type inequalities utilizing fractional integrals, J. Comput. Anal. Appl., 23 (2017), 1137–1145.
[10]	M. Ali, H. Kara, J. Tariboon, S. Asawasamrit, H. Budak, F. Hezenci, Some new Simpson's-formula-type inequalities for twice-differentiable convex functions via generalized fractional operators, Symmetry, 13 (2021), 2249. doi: 10.3390/sym13122249. doi: 10.3390/sym13122249
[11]	M. Vivas-Cortez, T. Abdeljawad, P. Mohammed, Y. Rangel-Oliveros, Simpson's integral inequalities for twice differentiable convex functions, Math. Probl. Eng., 2020 (2020), 1936461. doi: 10.1155/2020/1936461. doi: 10.1155/2020/1936461
[12]	T. Abdeljawad, S. Rashid, Z. Hammouch, İ. İşcan, Y. M. Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, Adv. Differ. Equ., 2020 (2020), 496. doi: 10.1186/s13662-020-02955-9. doi: 10.1186/s13662-020-02955-9
[13]	S. Butt, A. Akdemir, M. Bhatti, M. Nadeem, New refinements of Chebyshev-Pólya-Szegö-type inequalities via generalized fractional integral operators, J. Inequal. Appl., 2020 (2020), 157. doi: 10.1186/s13660-020-02425-6. doi: 10.1186/s13660-020-02425-6
[14]	S. Butt, E. Set, S. Yousaf, T. Abdeljawad, W. Shatanawi, Generalized integral inequalities for ABK-fractional integral operators, AIMS Mathematics, 6 (2021), 10164–10191. doi: 10.3934/math.2021589. doi: 10.3934/math.2021589
[15]	S. Butt, S. Yousaf, A. Asghar, K. Khan, H. Moradi, New Fractional Hermite-Hadamard-Mercer Inequalities for Harmonically Convex Function, J. Funct. Space., 2021 (2021), 5868326. doi:10.1155/2021/5868326. doi: 10.1155/2021/5868326
[16]	F. Ertuǧral, M. Sarikaya, Simpson type integral inequalities for generalized fractional integral, RACSAM, 113 (2019), 3115–3124. doi: 10.1007/s13398-019-00680-x. doi: 10.1007/s13398-019-00680-x
[17]	S. Hussain, J. Khalid, Y. Chu, Some generalized fractional integral Simpson's type inequalities with applications, AIMS Mathematics, 5 (2020), 5859–5883. doi: 10.3934/math.2020375. doi: 10.3934/math.2020375
[18]	A. Kashuri, B. Meftah, P. Mohammed, Some weighted Simpson type inequalities for differentiable $ s $-convex functions and their applications, Journal of Fractional Calculus and Nonlinear Systems, 1 (2021), 75–94. doi: 10.48185/jfcns.v1i1.150. doi: 10.48185/jfcns.v1i1.150
[19]	A. Kashuri, P. Mohammed, T. Abdeljawad, F. Hamasalh, Y. Chu, New Simpson type integral inequalities for $ s $-convex functions and their applications, Math. Probl. Eng., 2020 (2020), 8871988. doi: 10.1155/2020/8871988. doi: 10.1155/2020/8871988
[20]	S. Kermausuor, Simpson's type inequalities via the Katugampola fractional integrals for s-convex functions, Kragujev. J. Math., 45 (2021), 709–720.
[21]	C. Luo, T. Du, Generalized Simpson type inequalities involving Riemann-Liouville fractional integrals and their applications, Filomat, 34 (2020), 751–760. doi: 10.2298/FIL2003751L. doi: 10.2298/FIL2003751L
[22]	S. Rashid, A. Akdemir, F. Jarad, M. Noor, K. Noor, Simpson's type integral inequalities for $\mathit{\kappa }$-fractional integrals and their applications, AIMS Mathematics, 4 (2019), 1087–1100. doi: 10.3934/math.2019.4.1087. doi: 10.3934/math.2019.4.1087
[23]	M. Sarıkaya, H. Budak, S. Erden, On new inequalities of Simpson's type for generalized convex functions, Korean J. Math., 27 (2019), 279–295. doi: 10.11568/kjm.2019.27.2.279. doi: 10.11568/kjm.2019.27.2.279
[24]	E. Set, A. Akdemir, M. Özdemir, Simpson type integral inequalities for convex functions via Riemann-Liouville integrals, Filomat, 31 (2017), 4415–4420. doi: 10.2298/FIL1714415S. doi: 10.2298/FIL1714415S
[25]	H. Lei, G. Hu, J. Nie, T. Du, Generalized Simpson-type inequalities considering first derivatives through the $ \mathit{k}$-Fractional Integrals, IJAM, 50 (2020), 1–8.
[26]	H. Budak, S. Erden, M. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Meth. Appl. Sci., 44 (2021), 378–390. doi: 10.1002/mma.6742. doi: 10.1002/mma.6742
[27]	J. Hua, B. Y. Xi, F. Qi, Some new inequalities of Simpson type for strongly $s$-convex functions, Afr. Mat., 26 (2015), 741–752. doi: 10.1007/s13370-014-0242-2. doi: 10.1007/s13370-014-0242-2
[28]	S. Hussain, S. Qaisar, More results on Simpson's type inequality through convexity for twice differentiable continuous mappings, SpringerPlus, 5 (2016), 77. doi: 10.1186/s40064-016-1683-x. doi: 10.1186/s40064-016-1683-x
[29]	Y. Li, T. Du, Some Simpson type integral inequalities for functions whose third derivatives are ($\alpha \mathit{, m}$)-GA-convex functions, J. Egypt. Math. Soc., 24 (2016), 175–180. doi: 10.1016/j.joems.2015.05.009. doi: 10.1016/j.joems.2015.05.009
[30]	Z. Liu, An inequality of Simpson type, Proc. R. Soc. A, 461 (2005), 2155–2158. doi: 10.1098/rspa.2005.1505. doi: 10.1098/rspa.2005.1505
[31]	W. Liu, Some Simpson type inequalities for $h$-convex and ($\alpha \mathit{, m}$)-convex functions, J. Comput. Anal. Appl., 16 (2014), 1005–1012.
[32]	S. S. Dragomir, R. Agarwal, P. Cerone, On Simpson's inequality and applications, J. Inequal. Appl., 5 (2000), 533–579. doi: 10.1155/S102558340000031X. doi: 10.1155/S102558340000031X
[33]	M. Sarikaya, E. Set, M. Özdemir, On new inequalities of Simpson's type for functions whose second derivatives absolute values are convex, J. Appl. Math. Stat. Inf., 9 (2013), 37–45.
[34]	H. Budak, H. Kara, F. Hezenci, Fractional Simpson type inequalities for twice differentiable functions, submitted for publication.
[35]	F. Hezenci, H. Budak, H. Kara, New version of Fractional Simpson type inequalities for twice differentiable functions, Adv. Differ. Equ., 2021 (2021), 460. doi: 10.1186/s13662-021-03615-2. doi: 10.1186/s13662-021-03615-2
[36]	M. Sarikaya, F. Ertugral, On the generalized Hermite-Hadamard inequalities, Ann. Univ. Craiova-Mat., 47 (2020), 193–213.
[37]	A. Kashuri, E. Set, R. Liko, Some new fractional trapezium-type inequalities for preinvex functions, Fractal Fract., 3 (2019), 12. doi: 10.3390/fractalfract3010012. doi: 10.3390/fractalfract3010012
[38]	H. Budak, F. Ertuǧral, E. Pehlivan, Hermite-Hadamard type inequalities for twice differantiable functions via generalized fractional integrals, Filomat, 33 (2019), 4967–4979. doi: 10.2298/FIL1915967B. doi: 10.2298/FIL1915967B
[39]	H. Budak, E. Pehlivan, P. Kösem, On new extensions of Hermite-Hadamard inequalities for generalized fractional integrals, Communications in Mathematical Analysis, 18 (2021), 73–88. doi: 10.22130/SCMA.2020.121963.759. doi: 10.22130/SCMA.2020.121963.759
[40]	H. Budak, S. Yildirim, H. Kara, H. Yildirim, On new generalized inequalities with some parameters for coordinated convex functions via generalized fractional integrals, Math. Meth. Appl. Sci., 44 (2021), 13069–13098. doi: 10.1002/mma.7610. doi: 10.1002/mma.7610
[41]	P. Mohammed, M. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math., 372 (2020), 112740. doi: 10.1016/j.cam.2020.112740. doi: 10.1016/j.cam.2020.112740
[42]	X. You, M. Ali, H. Budak, P. Agarwal, Y. Chu, Extensions of Hermite–Hadamard inequalities for harmonically convex functions via generalized fractional integrals, J. Inequl. Appl., 2021 (2021), 102. doi: 10.1186/s13660-021-02638-3. doi: 10.1186/s13660-021-02638-3
[43]	D. Zhao, M. Ali, A. Kashuri, H. Budak, M. Sarikaya, Hermite–Hadamard-type inequalities for the interval-valued approximately $h$-convex functions via generalized fractional integrals, J. Inequal. Appl., 2020 (2020), 222. doi: 10.1186/s13660-020-02488-5. doi: 10.1186/s13660-020-02488-5

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