Riemann-Liouville fractional trapezium-like inequalities via generalized (m, h₁, h₂)-preinvexity.
Keywords:
Hermite-Hadamard’s inequality, Fractional integrals, Generalized (m, h₁, h₂)-preinvex functionsAbstract
In this paper, we derive a fractional integral identity concerning three times differentiable generalized preinvex mappings defined on minvex set. By using of this identity, we obtain new estimates on generalization of trapezium-like inequalities for functions whose third order derivatives are generalized (m, h₁, h₂)-preinvex via Riemann-Liouville fractional integrals. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.
References
G. A. Anastassiou, Generalised fractional Hermite-Hadamard inequalities involving m-convexity and (s, m)—convexity, Facta Univ. Ser. Math. Inform., 28, No. 2, pp. 107-126, (2013).
M. U. Awan, M. A. Noor, M. V. Mihai, K. I. Noor, Two point trapezoidal like inequalities involving hypergeometric functions, Filomat, 31, No. 8, pp. 2281-2292, (2017).
M. U. Awan, M. A. Noor, M. V. Mihai, K. I. Noor, Fractional Hermite-Hadamard inequalities for differentiable s-Godunova-Levin functions, Filomat, 30, No. 12, pp. 3235-3241, (2016).
F. X. Chen, Extensions of the Hermite-Hadamard inequality for convex functions via fractional integrals, J. Math. Inequal., 10, No. 1, 75-81, (2016).
F. X. Chen, Extensions of the Hermite-Hadamard inequality for harmonically convex functions via fractional integrals, Appl. Math. Comput., 268, pp. 121-128, (2015).
S. S. Dragomir, M. I. Bhatti, M. Iqbal, M. Muddassar, Some new Hermite-Hadamard’s type fractional integral inequalities, J. Comput. Anal. Appl., 18, No. 4, pp. 655-661, (2015).
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, pp. 3112-3126, (2016).
T. S. Du, J. G. Liao, L. Z. Chen, M. U. Awan, Properties and Riemann-Liouville fractional Hermite-Hadamard inequalities for the generalized (α, m)-preinvex functions, J. Inequal. Appl., 2016, Article No. 306, 24 pages, (2016).
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, pp. 358-369, (2017).
S.-R. Hwang, S.-Y. Yeh, K.-L. Tseng, Refinements and similar extensions of Hermite-Hadamard inequality for fractional integrals and their applications, Appl. Math. Comput., 249, pp. 103-113, (2014).
S.-R. Hwang, K.-L. Tseng, K.-C. Hsu, New inequalities for fractional integrals and their applications, Turkish J. Math., 40, pp. 471-486, (2016).
M. Iqbal, M. I. Bhatti, K. Nazeer, Generalization of inequalities analogous to Hermite-Hadamard inequality via fractional integrals, Bull. Korean Math. Soc., 52, No. 3, pp. 707-716, (2015).
İ. İşcan, New general integral inequalities for quasi-geometrically convex functions via fractional integrals, J. Inequal. Appl., Article No. 491, 15 pages, (2013).
İ. İşcan, S. H. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238, pp. 237-244, (2014).
A. Kashuri, R. Liko, Generalizations of Hermite-Hadamard and Ostrowski type inequalities for MTm-preinvex functions, Proyecciones (Antofagasta), 36, No. 1, pp. 45-80, (2017).
M. A. Latif, On some new inequalities of Hermite-Hadamard type for functions whose derivatives are s-convex in the second sense in the absolute value, Ukrainian Math. J., 67, No. 10, pp. 1552-1571, (2016).
M. A. Latif, S. S. Dragomir, E. Momoniat, On Hermite-Hadamard type integral inequalities for n-times differentiable m- and (α; m)- logarithmically convex functions, Filomat, 30, No. 11, pp. 3101-3114, (2016).
M. A. Latif, S. S. Dragomir, Generalization of Hermite-Hadamard type inequalities for n-times differentiable functions through preinvexity, Georgian Math. J., 23, No.1, pp. 97-104, (2016).
Y. J. Li, T. S. Du, B. Yu, Some new integral inequalities of Hadamard-Simpson type for extended (s, m)-preinvex functions, Ital. J. Pure Appl. Math., 36, pp. 583-600, (2016).
Y. M. Liao, J. H. Deng, J. R. Wang, Riemann-Liouville fractional Hermite-Hadamard inequalities. Part II: for twice differentiable geometric-arithmetically s-convex functions, J. Inequal. Appl., 2013, Article No. 517, 13 pages, (2013).
M. Matłoka, Inequalities for h-preinvex functions, Appl. Math. Comput., 234, pp. 52-57, (2014).
M. Matłoka, Some inequalities of Hadamard type for mappings whose second derivatives are h-convex via fractional integrals, J. Fract. Calc. Appl., 6, No. 1, pp. 110-119, (2015).
M. A. Noor, K. I. Noor, M. U. Awan, New fractional estimates of Hermite-Hadamard inequalities and applications to means, Stud. Univ. Babe¸ s-Bolyai Math., 61, No. 1, pp. 3-15, (2016).
M. A. Noor, K. I. Noor, M. U. Awan, Fractional Hermite-Hadmard inequalities for convex functions and applications, Tbilisi Math. J., 8, No. 2, pp. 103-113, (2015).
M. A. Noor, K. I. Noor, M. V. Mihai, M. U. Awan, Fractional HermiteHadamard inequalities for some classes of differentiable preinvex functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 78, No. 3, pp. 163-174, (2016).
O. Omotoyinbo, A. Mogbademu, Some new Hermite-Hadamard integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1, No. 1, pp. 001-012, (2014).
M. E. Özdemir, S. S. Dragomir, Ǹ . Yildiz, The Hadamard inequality for convex function via fractional integrals, Acta Math. Sci. Ser. B Engl. Ed., 33B, No. 5, pp. 1293-1299, (2013).
M. E. Özdemir, A. Ekinci, Generalized integral inequalities for convex functions, Math. Inequal. Appl., 19, No. 4, pp. 1429-1439, (2016).
C. Peng, C. Zhou, T. S. Du, Riemann-Liouville fractional Simpson’s inequalities through generalized (m, h₁, h₂)-preinvexity, Ital. J. Pure Appl. Math., 38, pp. 345-367, (2017).
S. Qaisar, M. Iqbal, and M. Muddassar, New Hermite-Hadamard’s inequalities for preinvex functions via fractional integrals, J. Comput. Anal. Appl., 20, No. 7, pp. 1318-1328, (2016).
M. Z. Sarikaya, E. Set, H. Yaldiz, N. Ba¸ sak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57, pp. 2403-2407, (2013).
M. Z. Sarikaya, H. Budak, Generalized Hermite-Hadamard type integral inequalities for fractional integrals, Filomat, 30, No. 5, pp. 1315-1326, (2016).
M. Tunç, E. Göv, Ü. Şanal, On tgs-convex function and their inequalities, Facta Univ. Ser. Math. Inform., 30, No. 5, pp. 679-691, (2015).
J. R. Wang, X. Z. Li, M. Fečkan, Y. Zhou, Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92, No. 11, pp. 2241-2253, (2013).
J. Wang, J. Deng, M. Fečkan, Hermite-Hadamard-type inequalities for r-convex functions based on the use of Riemann-Liouville frantional integrals, Ukrainian Math. J., 65, No. 2, pp. 193-211, (2013).
T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl., 136, pp. 29-38, (1988).
S.-H. Wu, B. Sroysang, J.-S. Xie, and Y.-M. Chu, Parametrized inequality of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex, SpringerPlus, 4, Article No. 831, 9 pages, (2015).
B. -Y. Xi, F. Qi, Hermite-Hadamard type inequalities for geometrically r-convex functions, Studia Sci. Math. Hungar., 51, No. 4, pp. 530-546, (2014).
B. Y. Xi, S. H. Wang, and F. Qi, Some inequalities of Hermite-Hadamard type for functions whose 3rd derivatives are P -convex, Applied Mathematics, 3, pp. 1898-1902, (2012).
Y. R. Zhang, J. R. Wang, On some new Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, J. Inequal. Appl., Article No. 220, 27 pages, (2013).
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