Hermite-Hadamard type fractional integral inequalities for products of two MT(r;g,m,φ)-preinvex functions
DOI:
https://doi.org/10.22199/issn.0717-6279-2020-01-0014Keywords:
Hermite-Hadamard type inequality, Hölder’s inequality, Minkowski’s inequality, Cauchy’s inequality, Power mean inequality, Riemann-Liouville fractional integral, s-convex function in the second sense, m-invex, P -functionAbstract
A new class of MT(r;g,m,φ)-preinvex functions is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving products of two MT(r;g,m,φ)-preinvex functions are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for products of two MT(r;g,m,φ)-preinvex functions via Riemann-Liouville fractional integrals are established. These general inequalities give us some new estimates for the left-hand side of Gauss-Jacobi type quadrature formula and Hermite-Hadamard type fractional integral inequalities. At the end, some conclusions and future research are given.
References
A. Akkurt and H. Yildirim, “On some fractional integral inequalities of Hermite-Hadamard type for r-preinvex functions”, Khayyam journal of mathematics, vol. 2, no. 2, pp. 120-127, 2016, doi: 10.22034/kjm.2016.40640.
T. Antczak, “Mean value in invexity analysis”, Nonlinear analysis: theory, methods & applications, vol. 60, no. 8, pp. 1473-1484, Mar. 2005, doi: 10.1016/j.na.2004.11.005.
F. Chen, “A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals”, Italian journal of pure applied mathematics, no. 33, pp. 299-306, 2014. [On line]. Available: https://bit.ly/2vJP0QR
F. X. Chen and S. H. Wu, “Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions”, Journal of nonlinear sciences and applications, vol. 9, no. 2, pp. 705-716, 2016, doi: 10.22436/jnsa.009.02.32.
Y. M. Chu, M. A. Khan, T. U. Khan, and T. Ali, “Generalizations of Hermite-Hadamard type inequalities for MT -convex functions”, Journal of nonlinear sciences and applications, vol. 9, no. 6, pp. 4305-4316, 2016, doi: 10.22436/jnsa.009.06.72.
Y. M. Chu, G. D. Wang, and X. H. Zhang, “Schur convexity and Hadamard’s inequality”, Mathematical inequalities & applications, vol. 13, no. 4, pp. 725-731, 2010, doi: 10.7153/mia-13-51.
S. S. Dragomir, J. Pěcarić, and L. E. Persson, “Some inequalities of Hadamard type”, Soochow journal of mathematics, vol. 21, no. 3, pp. 335-341, Jul. 1995. [On line]. Available: https://bit.ly/37UvWO9
T. S. Du, J. G. Liao, and Y. J. Li, “Properties and integral inequalities of Hadamard-Simpson type for the generalized (s, m)-preinvex functions”, Journal of nonlinear sciences and applications, vol. 9, no. 5, pp. 3112-3126, 2016, doi: 10.22436/jnsa.009.05.102.
H. Hudzik and L. Maligranda, “Some remarks on s-convex functions”, Aequationes mathematicae, vol. 48, no. 1, pp. 100-111, Aug. 1994, doi: 10.1007/BF01837981.
W. D. Jiang, D. W. Niu, and F. Qi, “Some inequalties of Hermite-Hadamard type for r-φ-preinvex functions”, Tamkang journal of mathematics, vol. 45, no. 1, pp. 31-38, 2014, doi: 10.5556/j.tkjm.45.2014.1261.
H. Kavurmaci, M. Avci. and M. E. Özdemir, “New inequalities of Hermite-Hadamard type for convex functions with applications”, Jun. 2010, arXiv:1006.1593v1
M. A. Khan, Y. Khurshid, T. Ali, and N. Rehman,”Inequalities for three times differentiable functions”, The Punjab university journal of mathematics, vol. 48, no. 2, pp. 35-48, 2016. [On line]. Available: https://bit.ly/2vNZjU7
M. A. Khan, Y. Khurshid, and T. Ali, “Hermite-Hadamard inequality for fractional integrals via η-convex functions”, Acta mathematica universitatis comenianae, vol. 86, no. 1, pp. 153—164, 2017. [On line]. Available: https://bit.ly/2v3TEJm
W. Liu, “New integral inequalities involving beta function via P –convexity”, Miskolc mathematical notes (Online), vol. 15, no. 2, pp. 585-591, 2014, doi: 10.18514/mmn.2014.660.
W. Liu, W. Wen and J. Park, “Ostrowski type fractional integral inequalities for MT -convex functions”, Miskolc mathematical notes (Online), vol. 16, no. 1, pp. 249-256, 2015, doi: 10.18514/MMN.2015.1131.
W. Liu, W. Wen and J. Park, “Hermite-Hadamard type inequalities for MT -convex functions via classical integrals and fractional integrals”, Journal of nonlinear sciences and applications, vol. 9, no. 3, pp. 766-777, 2016, doi: 10.22436/jnsa.009.03.05.
M. E. Özdemir, E. Set, and M. Alomari, “Integral inequalities via several kinds of convexity”, Creative mathematics and informatics, vol. 20, no. 1, pp. 62-73, 2011. [On line]. Available: https://bit.ly/31rqeRw
R. Pini, “Invexity and generalized convexity”, Optimization, vol. 22, no. 4. pp. 513— 525, Jun. 1991, doi: 10.1080/02331939108843693.
F. Qi and B. Y. Xi, “Some integral inequalities of Simpson type for GA− ɛ-convex functions”, Georgian mathematical journal, vol. 20, no. 4, pp. 775-788, Oct. 2013, doi: 10.1515/gmj-2013-0043.
H. N. Shi, “Two Schur-convex functions related to Hadamard-type integral inequalities”, Publicationes mathematicae debrecen, vol. 78, no. 2, pp. 393-403, 2011. [On line]. Available: https://bit.ly/2v2X9PW
D. D. Stancu, G. Coman, O. Agratini, and Trîmbitaș Radu, Analiză numerică și teoria aproximării, vol. 2. Cluj-Napoca: Presa Universitară Clujeana, 2001.
M. Tunç and H. Yildirim, “On MT –convexity”, May 2012. arXiv:1205.5453v1
X. M. Yang, X. Q. Yang, and K. L. Teo, “Generalized invexity and generalized invariant monotonicity”, Journal of optimization theory and applications, vol. 117, no. 3, pp. 607-625, Jun. 2003, doi: 10.1023/A:1023953823177.
X. M. Zhang, Y. M. Chu and X. H. Zhang, “The Hermite-Hadamard type inequality of GA-convex functions and its applications”, Journal of inequalities and applications, Art. ID. 507560, Dec. 2010, doi: 10.1155/2010/507560.
Published
How to Cite
Issue
Section
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
