SciELO - Scientific Electronic Library Online

 
vol.37 issue2Solutions and stability of a variant of Wilson's functional equationOn velocity bimagnetic biharmonic particles with energy on Heisenberg space author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.37 no.2 Antofagasta June 2018

http://dx.doi.org/10.4067/S0716-09172018000200345 

Articles

Riemann-Liouville fractional trapezium-like inequalities via generalized (m, h 1 , h 2 )-preinvexity

Piao Guo1 

Zhengzheng Huang2 

Tingsong Du3 

1China Three Gorges University, College of Science, Department of Mathematics, Yichang 443002, Hubei, P. R. China. E-mail: guopiaoctgu@163.com

2China Three Gorges University, College of Science, Department of Mathematics, Yichang 443002, Hubei, P. R. China. E-mail: huangzhengctgu@163.com

3China Three Gorges University, College of Science, Department of Mathematics, Yichang 443002, Hubei, P. R. China. E-mail: tingsongdu@ctgu.edu.cn

Abstract:

In this paper, we derive a fractional integral identity concerning three times differentiable generalized preinvex mappings defined on m-invex 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.

Key words and phrases : Hermite-Hadamard's inequality; fractional integrals; generalized (m, h1, h2)-preinvex functions.

2010 Mathematics Subject Classification: Primary 26A33; 26A51; Secondary 26D07, 26D20, 41A55.

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 61374028).

References.

[1] G. A. Anastassiou, Generalised fractional Hermite-Hadamard inequalities involving m-convexity and (s,m)-convexity, Facta Univ. Ser. Math. Inform., 28, N°. 2, pp. 107-126, (2013). [ Links ]

[2] M. U. Awan, M. A. Noor, M. V. Mihai, K. I. Noor, Two point trapezoidal like inequalities involving hypergeometric functions, Filomat, 31, N°. 8, pp. 2281-2292, (2017). [ Links ]

[3] M. U. Awan, M. A. Noor, M. V. Mihai, K. I. Noor, Fractional Hermite-Hadamard inequalities for differentiable s-Godunova-Levin functions, Filomat, 30, N°. 12, pp. 3235-3241, (2016). [ Links ]

[4] F. X. Chen, Extensions of the Hermite-Hadamard inequality for convex functions via fractional integrals, J. Math. Inequal., 10, No. 1, 75-81, (2016). [ Links ]

[5] F. X. Chen, Extensions of the Hermite-Hadamard inequality for harmonically convex functions via fractional integrals, Appl. Math. Comput., 268, pp. 121-128, (2015). [ Links ]

[6] 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). [ Links ]

[7] 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). [ Links ]

[8] 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). [ Links ]

[9] 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). [ Links ]

[10] 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). [ Links ]

[11] 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). [ Links ]

[12] 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). [ Links ]

[13] İ. İşcan, New general integral inequalities for quasi-geometrically convex functions via fractional integrals, J. Inequal. Appl., Article N°. 491, 15 pages, (2013). [ Links ]

[14] İ. İşcan, S. H. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238, pp. 237-244, (2014). [ Links ]

[15] A. Kashuri, R. Liko, Generalizations of Hermite-Hadamard and Ostrowski type inequalities for MTm-preinvex functions, Proyecciones (Antofagasta), 36, N°. 1, pp. 45-80, (2017). [ Links ]

[16] 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, N°. 10, pp. 1552-1571, (2016). [ Links ]

[17] 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, N°. 11, pp. 3101-3114, (2016). [ Links ]

[18] 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). [ Links ]

[19] 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). [ Links ]

[20] 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). [ Links ]

[21] M. Matłoka, Inequalities for h-preinvex functions, Appl. Math. Comput., 234, pp. 52-57, (2014). [ Links ]

[22] 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). [ Links ]

[23] M. A. Noor, K. I. Noor, M. U. Awan, New fractional estimates of Hermite-Hadamard inequalities and applications to means, Stud. Univ. Babeş-Bolyai Math., 61, No. 1, pp. 3-15, (2016). [ Links ]

[24] 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). [ Links ]

[25] M. A. Noor, K. I. Noor, M. V. Mihai, M. U. Awan, Fractional Hermite-Hadamard 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). [ Links ]

[26] 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). [ Links ]

[27] M. E. Özdemir, S. S. Dragomir, Ç. Yildiz, The Hadamard inequality for convex function via fractional integrals, Acta Math. Sci. Ser. B Engl. Ed., 33B, No. 5, pp. 1293-1299, (2013). [ Links ]

[28] M. E. Özdemir, A. Ekinci, Generalized integral inequalities for convex functions, Math. Inequal. Appl., 19, No. 4, pp. 1429-1439, (2016). [ Links ]

[29] 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). [ Links ]

[30] 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). [ Links ]

[31] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57, pp. 2403-2407, (2013). [ Links ]

[32] M. Z. Sarikaya, H. Budak, Generalized Hermite-Hadamard type integral inequalities for fractional integrals, Filomat, 30, No. 5, pp. 1315-1326, (2016). [ Links ]

[33] 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). [ Links ]

[34] 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). [ Links ]

[35] 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). [ Links ]

[36] T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl., 136, pp. 29-38, (1988). [ Links ]

[37] 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). [ Links ]

[38] 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). [ Links ]

[39] 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). [ Links ]

[40] 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). [ Links ]

Received: June 2017; Accepted: April 2018

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License