Estimates of trapezium-type inequalities for $ h $-convex functions with applications to quadrature formulae

Muhammad Samraiz; Fakhra Nawaz; Bahaaeldin Abdalla; Thabet Abdeljawad; Gauhar Rahman; Sajid Iqbal; Muhammad Samraiz; Fakhra Nawaz; Bahaaeldin Abdalla; Thabet Abdeljawad; Gauhar Rahman; Sajid Iqbal

doi:10.3934/math.2021443

AIMS Mathematics

2021, Volume 6, Issue 7: 7625-7648. doi: 10.3934/math.2021443

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Research article

Estimates of trapezium-type inequalities for $ h $-convex functions with applications to quadrature formulae

1.
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
2.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 12345, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4.
Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan
5.
Department of Mathematics and Statistics, Hazara University Mansehra, Pakistan
6.
Department of Mathematics, Ripha International University, Faisalabad Campus, Satyana Road, 38000 Faisalabad, Pakistan

Received: 19 December 2020 Accepted: 29 April 2021 Published: 10 May 2021
MSC : 26D15, 26D10, 26A33

In this article, we develop a new class of trapezium-type inequalities up to twice differentiable $ h $-convex mappings for fractional integrals of Riemann-type. We conclude numerous existing results in literature from our general inequalities. Based on our consequences, we will obtain some quadrature formulas as applications.
- mid-point type inequalities,
- generalized Riemann-Liouville fractional integral operator,
- $ h $-convex,
- $ s $-convex
Citation: Muhammad Samraiz, Fakhra Nawaz, Bahaaeldin Abdalla, Thabet Abdeljawad, Gauhar Rahman, Sajid Iqbal. Estimates of trapezium-type inequalities for $ h $-convex functions with applications to quadrature formulae[J]. AIMS Mathematics, 2021, 6(7): 7625-7648. doi: 10.3934/math.2021443

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Abstract

In this article, we develop a new class of trapezium-type inequalities up to twice differentiable $ h $-convex mappings for fractional integrals of Riemann-type. We conclude numerous existing results in literature from our general inequalities. Based on our consequences, we will obtain some quadrature formulas as applications.

References

[1]	J. Liao, S. Wu, T. Du, The Sugeno integral with respect to $\alpha$-preinvex functions, Fuzzy Set. Syst., 379 (2020), 102-114. doi: 10.1016/j.fss.2018.11.008
[2]	S. Wu, M. U. Awan, Estimates of upper bound for a function associated with Riemann-Liouville fractional integral via $h$-convex functions, J. Funct. Space., 2019 (2019), 1-7.
[3]	İ. İşcan, S. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237-244.
[4]	S. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, Rocky Mt. J. Math., 39 (2009), 1741-1749.
[5]	Y. Bai, S. Wu, Y. Wu, Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated $(s, m)$-$P$-convex, J. Funct. Space., 2018 (2018), 1693075.
[6]	S. Wu, I. A. Baloch, İ. İşcan, On Harmonically $(p, h, m)$-preinvex functions, J. Funct. Space., 2017 (2017), 2148529.
[7]	G. Toader, Some generalizations of the convexity, Univ. Cluj-Napoca, Cluj-Napoca, (1985), 329-338.
[8]	C. P. Niculescu, L. E. Persson, Convex functions and their applications: A contemporary approach, CMC Books in Mathematics, New York, USA, 2004.
[9]	S. Varosanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), 303-311.
[10]	J. Pečarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings and statistical application, Acadmic Press, New York, USA, 1992.
[11]	I. Koca, P. Yaprakdal, A new approach for nuclear family model with fractional order Caputo derivative, Appl. Math. Nonlinear Sci., 5 (2020), 393-404. doi: 10.2478/amns.2020.1.00037
[12]	S. Kabra, H. Nagar, K. S. Nisar, D. L. Suthar, The Marichev-Saigo-Maeda fractional calculus operators pertaining to the generalized k-Struve function, Appl. Math. Nonlinear Sci., 5 (2020), 593-602. doi: 10.2478/amns.2020.2.00064
[13]	M. E. Özdemir, M. Avci, E. Set, On some inequalities of Hermite-Hadamard-type via $m$-convexity, Appl. Math. Lett., 23 (2010), 1065-1070. doi: 10.1016/j.aml.2010.04.037
[14]	X. Wu, J. Wang, J. Zhang, Hermite-Hadamard-type inequalities for convex functions via the fractional integrals with exponential kernel, Mathematics, 7 (2019), 845. doi: 10.3390/math7090845
[15]	S. Rashid, T. Abdeljawad, F. Jarad, M. A. Noor, Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications, Mathematics, 7 (2019), 807. doi: 10.3390/math7090807
[16]	E. Set, M. E. Özdemir, S. S. Dragomir, On the Hadamard-type inequalities involving several kinds of convexity, J. Inequal. Appl., 2010 (2010), 286845. doi: 10.1155/2010/286845
[17]	E. Set, M. E. Özdemir, S. S. Dragomir, On the Hermite-Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl., 2010 (2010), 148102. doi: 10.1155/2010/148102
[18]	M. Z. Sarikaya, Y. Hüseyin, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17 (2017), 1049-1059. doi: 10.18514/MMN.2017.1197
[19]	Y. Zhang, J. Wang, On some new Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, J. Inequal. Appl., 2013 (2013), 220. doi: 10.1186/1029-242X-2013-220
[20]	S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10 (2009), 86.
[21]	M. Z. Sarikaya, N. Aktan, On the generalizations of some integral inequalitirs and their applications, Math. Comput. Model., 54 (2011), 2175-2182. doi: 10.1016/j.mcm.2011.05.026
[22]	M. E. Özdemir, M. Avci, H. Kavurmaci, Hermite-Hadamard-type inequalities via $(\alpha, m)$-convexity, Comput. Math. Appl., 61 (2011), 2614-2620. doi: 10.1016/j.camwa.2011.02.053
[23]	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. doi: 10.1016/j.mcm.2011.12.048
[24]	Y. Liao, J. Deng, J. Wang, Riemann-Liouville fractional Hermite-Hadamard inequalities. Part II: For twice differentiable geometric-arithmetically s-convex functions, J. Inequal. Appl., 2013 (2013), 1-13. doi: 10.1186/1029-242X-2013-1
[25]	S. Wu, S. Iqbal, M. Aamir, M. Samraiz, A. Younus, On some Hermite-Hadamard inequalities involving $k$-fractional calculus, J. Inequal. Appl., 2021 (2021), 32. doi: 10.1186/s13660-020-02527-1
[26]	S. Iqbal, M. Aamir, M. Samraiz, Fractional Hermite-Hadamard inequalities for twice differenciable geometric-arithmetically $s$-convex functions, J. Math. Anal., 11 (2020), 13-31.
[27]	M. Tunc, On new inequalities for h-convex functions via Riemann Liouville fractional integration, Filomat, 27 (2013), 559-565. doi: 10.2298/FIL1304559T
[28]	S. S. Dragomir, J. E. Pečaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.
[29]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, New York, London, 2006.
[30]	J. Wang, X. Li, M. Fečkan, Y. Zhou, Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92 (2012), 2241-2253.
[31]	J. Deng, J. Wang, Fractional Hermite-Hadamard inequalities for $(\alpha, m)$-logrithmically convex functions, J. Inequali. Appl., 2013 (2013), 364. doi: 10.1186/1029-242X-2013-364
[32]	S. Mubeen, G. M. Habibullah, $k$-Fractional integrals and application, Int. J. Contemp. Math. Sci., 7 (2016), 89-94.

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