Some parameterized integral inequalities for <em>p</em>-convex mappings via the right Katugampola fractional integrals

Gou Hu; Hui Lei; Tingsong Du; Gou Hu; Hui Lei; Tingsong Du

doi:10.3934/math.2020098

AIMS Mathematics

2020, Volume 5, Issue 2: 1425-1445. doi: 10.3934/math.2020098

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Research article

Some parameterized integral inequalities for p-convex mappings via the right Katugampola fractional integrals

Gou Hu ¹,
Hui Lei ²,
Tingsong Du ^{2
,
,}

1.
School of Mathematics, Hunan University, Changsha 410082, P. R. China
2.
Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, P. R. China

Received: 07 November 2019 Accepted: 09 January 2020 Published: 20 January 2020
MSC : 26A33, 26A51, 26D15, 26D10

We use the definition of a fractional integral operators, proposed by Katugampola, to establish a fractional Hermite-Hadamard's inequality for p-convex mappings and an identity with two parameters. We derive several parameterized integral inequalities associated with this identity, and provide three examples to illustrate the obtained results.
- Hermite-Hadamard’s inequality,
- Simpson’s inequality,
- Riemann-Liouville fractional integrals,
- Hadamard fractional integrals
Citation: Gou Hu, Hui Lei, Tingsong Du. Some parameterized integral inequalities for p-convex mappings via the right Katugampola fractional integrals[J]. AIMS Mathematics, 2020, 5(2): 1425-1445. doi: 10.3934/math.2020098

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Abstract

We use the definition of a fractional integral operators, proposed by Katugampola, to establish a fractional Hermite-Hadamard's inequality for p-convex mappings and an identity with two parameters. We derive several parameterized integral inequalities associated with this identity, and provide three examples to illustrate the obtained results.

References

[1]	P. Agarwal, Some inequalities involving Hadamard-type k-fractional integral operators, Math. Meth. Appl. Sci., 40 (2017), 3882-3891. doi: 10.1002/mma.4270
[2]	M. U. Awan, M. A. Noor, T. S. Du, et al. New refinements of fractional Hermite-Hadamard inequality, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math., 113 (2019), 21-29.
[3]	H. Chen, U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274-1291. doi: 10.1016/j.jmaa.2016.09.018
[4]	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.
[5]	T. S. Du, M. U. Awan, A. Kashuri, et al. Some k-fractional extensions of the trapezium inequalities through generalized relative semi-(m, h)-preinvexity, Appl. Anal., 2019 (2019), 1-21.
[6]	T. S. Du, J. G. Liao, Y. J. Li, Properties and integral inequalities of Hadamard-Simpson type for the generalized (s, m)-preinvex functions, J. Nonlinear Sci. Appl., 9 (2016), 3112-3126. doi: 10.22436/jnsa.009.05.102
[7]	T. S. Du, Y. J. Li, Z. Q. Yang, A generalization of Simpson's inequality via differentiable mapping using extended (s, m)-convex functions, Appl. Math. Comput., 293 (2017), 358-369.
[8]	R. S. Dubey, P. Goswami, Some fractional integral inequalities for the Katugampola integral operator, AIMS Mathematics, 4 (2019), 193-198. doi: 10.3934/math.2019.2.193
[9]	G. Farid, W. Nazeer, M. S. Saleem, et al. Bounds of Riemann-Liouville fractional integrals in general form via convex functions and their applications, Mathematics, 6 (2018), 1-10.
[10]	M. GÜrbÜz, Y. Taşdan, E. Set, Ostrowski type inequalities via the Katugampola fractional integrals, AIMS Mathematics, 5 (2020), 42-53. doi: 10.3934/math.2020004
[11]	İ. İşcan, Ostrowski type inequalities for p-convex functions, New Trends Math. Sci., 4 (2016), 140-150. doi: 10.20852/ntmsci.2016318838
[12]	İ. İşcan, S. Turhan, S. Maden, Hermite-Hadamard and Simpson-like type inequalities for differentiable p-quasi-convex functions, Filomat, 31 (2017), 5945-5953. doi: 10.2298/FIL1719945I
[13]	M. Jleli, D. O'Regan, B. Samet, On Hermite-Hadamard type inequalities via generalized fractional integrals, Turkish J. Math., 40 (2016), 1221-1230. doi: 10.3906/mat-1507-79
[14]	U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.
[15]	S. Kermausuor, Generalized Ostrowski-type inequalities involving second derivatives via the Katugampola fractional integrals, J. Nonlinear Sci. Appl., 12 (2019), 509-522. doi: 10.22436/jnsa.012.08.02
[16]	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.
[17]	P. Kórus, An extension of the Hermite-Hadamard inequality for convex and s-convex functions, Aequationes Math., 93 (2019), 527-534. doi: 10.1007/s00010-019-00642-z
[18]	M. Kunt, İ. İşcan, Hermite-Hadamard-Fejér type inequalities for p-convex functions, Arab J. Math. Sci., 23 (2017), 215-230.
[19]	M. Kunt, İ. İşcan, Hermite-Hadamard type inequalities for p-convex functions via fractional integrals, Moroccan J. Pure and Appl. Anal., 3 (2017), 22-35. doi: 10.1515/mjpaa-2017-0003
[20]	M. Kunt, D. Karapınar, S. Turhan, et al. The right Rieaman-Liouville fractional Hermite- Hadamard type inequalities for convex functions, J. Inequal. Spec. Funct., 9 (2018), 45-57.
[21]	J. Liao, S. H. Wu, T. S. Du, The Sugeno integral with respect to α-preinvex functions, Fuzzy Sets and Systems, 379 (2020), 102-114. doi: 10.1016/j.fss.2018.11.008
[22]	N. I. Mahmudov, S. Emin, Fractional-order boundary value problems with Katugampola fractional integral conditions, Adv. Differ. Equ., 2018 (2018), 81.
[23]	M. Matłoka, Weighted Simpson type inequalities for h-convex functions, J. Nonlinear Sci. Appl., 10 (2017), 5770-5780. doi: 10.22436/jnsa.010.11.15
[24]	N. Mehreen, M. Anwar, Integral inequalities for some convex functions via generalized fractional integrals, J. Inequal. Appl., 2018 (2018), 208.
[25]	M. V. Mihai, M. U. Awan, M. A. Noor, et al. Fractional Hermite-Hadamard inequalities containing generalized Mittag-Leffler function, J. Inequal. Appl., 2017 (2017), 265.
[26]	İ. Mumcu, E. Set, A. O. Akdemir, Hermite-Hadamard type inequalities for harmonically convex functions via Katugampola fractional integrals, Miskolc Math. Notes, 20 (2019), 409-424. doi: 10.18514/MMN.2019.2722
[27]	M. A. Noor, K. I. Noor, S. Iftikhar, Nonconvex functions and integral inequalities, Punjab Univ. J. Math., 47 (2015), 19-27.
[28]	M. A. Noor, M. U. Awan, M. V. Mihai, et al. Hermite-Hadamard inequalities for differentiable p-convex functions using hypergeometric functions, Publications de L'Institut Mathématique (Beograd), 100 (2016), 251-257.
[29]	M. A. Noor, M. U. Awan, M. V. Mihai, et al. Bounds involving Gauss's hypergeometric functions via (p, h)-convexity, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2017), 41-48.
[30]	A. W. Roberts, D. E. Varberg, Convex Functions, Academic Press, New York, (1973).
[31]	M. Z. Sarikaya, E. Set, H. Yaldiz, et al. Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, J. Math. Comput. Model., 57 (2013), 2403-2407. doi: 10.1016/j.mcm.2011.12.048
[32]	E. Set, İ. İşcan, H. H. Kara, Hermite-Hadamard-Fejér type inequalities for s-convex function in the second sense via fractional integrals, Filomat, 30 (2016), 3131-3138. doi: 10.2298/FIL1612131S
[33]	A. Thatsatian, S. K. Ntouyas, J. Tariboon, Some Ostrowski type inequalities for p-convex functions via generalized fractional integrals, J. Math. Inequal., 13 (2019), 467-478.
[34]	T. Toplu, E. Set, İ. İşcan, et al. Hermite-Hadamard type inequalities for p-convex functions via Katugampola fractional integrals, Facta Univ. Ser. Math. Inform., 34 (2019), 149-164.
[35]	J. Vanterler da C. Sousa, E. Capelas de Oliveira, The Minkowski's inequality by means of a generalized fractional integral, AIMS Mathematics, 3 (2018), 131-147. doi: 10.3934/Math.2018.1.131
[36]	J. R. Wang, J. H. Deng, M. Fečkan, Exploring s-e-condition and applications to some Ostrowski type inequalities via Hadamard fractional integrals, Math. Slovaca, 64 (2014), 1381-1396.
[37]	S. H. Wu, M. U. Awan, Estimates of upper bound for a function associated with Riemann-Liouville fractional integral via h-convex functions, J. Funct. Spaces, 2019 (2019), 1-7.
[38]	S. D. Zeng, D. Baleanu, Y. R. Bai, et al. Fractional differential equations of Caputo-Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549-554.
[39]	K. S. Zhang, J. P. Wan, p-convex functions and their properties, Pure Appl. Math., 23 (2007), 130-133.

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