Some fractional integral inequalities via $ h $-Godunova-Levin preinvex function

Sabila Ali; Rana Safdar Ali; Miguel Vivas-Cortez; Shahid Mubeen; Gauhar Rahman; Kottakkaran Sooppy Nisar; Sabila Ali; Rana Safdar Ali; Miguel Vivas-Cortez; Shahid Mubeen; Gauhar Rahman; Kottakkaran Sooppy Nisar

doi:10.3934/math.2022763

AIMS Mathematics

2022, Volume 7, Issue 8: 13832-13844. doi: 10.3934/math.2022763

Previous Article Next Article

Research article

Some fractional integral inequalities via $ h $-Godunova-Levin preinvex function

1.
Department of Mathematics, University of Lahore, Lahore, Pakistan
2.
Escuela de Ciencias Físicas y Matemáticas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Católica del Ecuador, Av. 12 de octubre 1076, Apartado, Quito 17-01-2184, Ecuador
3.
Department of Mathematics, University of Sargodha P.O. Box 40100, Sargodha, Pakistan
4.
Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
5.
Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, 11991, Saudi Arabia

Received: 01 February 2022 Revised: 17 March 2022 Accepted: 28 March 2022 Published: 23 May 2022
MSC : 26D10, 26D15, 26D10, 26D53, 05A30

In recent years, integral inequalities are investigated due to their extensive applications in several domains. The aim of the paper is to investigate certain new fractional integral inequalities which include Hermite-Hadamard inequality and different forms of trapezoid type inequalities related to Hermite-Hadamard inequality for $ h $-Godunova-Levin preinvex function. Moreover, we compare our obtained results with the existing work in the literature and are represented by corollaries.
- fractional inequalities,
- $ h $-Godunova-Levin convex and preinvex function,
- Hadamard inequality
Citation: Sabila Ali, Rana Safdar Ali, Miguel Vivas-Cortez, Shahid Mubeen, Gauhar Rahman, Kottakkaran Sooppy Nisar. Some fractional integral inequalities via $ h $-Godunova-Levin preinvex function[J]. AIMS Mathematics, 2022, 7(8): 13832-13844. doi: 10.3934/math.2022763

Related Papers:

Abstract

In recent years, integral inequalities are investigated due to their extensive applications in several domains. The aim of the paper is to investigate certain new fractional integral inequalities which include Hermite-Hadamard inequality and different forms of trapezoid type inequalities related to Hermite-Hadamard inequality for $ h $-Godunova-Levin preinvex function. Moreover, we compare our obtained results with the existing work in the literature and are represented by corollaries.

References

[1]	C. J. Huang, G. Rahman, K. S. Nisar, A. Ghaffar, F. Qi, Some inequalities of Hermite-Hadamard type for $k$-fractional conformable integrals, Australian J. Math. Anal. Appl., 16 (2019), 1–9.
[2]	K. S. Nisar, G. Rahman, K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 2019 (2019), 245. https://doi.org/10.1186/s13660-019-2197-1 doi: 10.1186/s13660-019-2197-1
[3]	K.S. Niasr, A. Tassadiq, G. Rahman, A. Khan, Some inequalities via fractional conformable integral operators, J. Inequal. Appl., 2019 (2019), 217. https://doi.org/10.1186/s13660-019-2170-z doi: 10.1186/s13660-019-2170-z
[4]	F. Qi, G. Rahman, S. M. Hussain, W. S. Du, K. S. Nisar, Some inequalities of Čebyšev type for conformable $k$-fractional integral operators, Symmetry, 2018 (2018), 614. https://doi.org/10.3390/sym10110614 doi: 10.3390/sym10110614
[5]	G. Rahman, K. S. Nisar, F. Qi, Some new inequalities of the Gruss type for conformable fractional integrals, AIMS Math., 3 (2018), 575–583.
[6]	G. Rahman, K. S. Nisar, A. Ghaffar, F. Qi, Some inequalities of the Grüss type for conformable $k$-fractional integral operators, RACSAM, 114 (2020), 9. https://doi.org/10.1007/s13398-019-00731-3 doi: 10.1007/s13398-019-00731-3
[7]	G. Rahman, Z. Ullah, A. Khan, E. Set, K. S. Nisar, Certain Chebyshev type inequalities involving fractional conformable integral operators, Mathematics, 7 (2019), 364. https://doi.org/10.3390/math7040364 doi: 10.3390/math7040364
[8]	G. Rahmnan, T. Abdeljawad, F. Jarad, K. S. Nisar, Bounds of generalized proportional fractional integrals in general form via convex functions and their applications, Mathematics, 8 (2020), 113. https://doi.org/10.3390/math8010113 doi: 10.3390/math8010113
[9]	X. Z. Yang, G. Farid, W. Nazeer, Y. M. Chu, C. F. Dong, Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex function, AIMS Math., 5 (2020), 6325–6340.
[10]	M. Vivas-Cortez, M. A. Ali, A. Kashuri, H. Budak, A. Vlora, Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions, AIMS Math, 6 (2021), 9397–9421. https://doi.org/10.3934/math.2021546 doi: 10.3934/math.2021546
[11]	A. Guessab, Sharp Approximations Based on Delaunay Triangulations and Voronoi Diagrams, NSU Publishing and Printing center, 2022,386.
[12]	L. ER, Uber die fourierreihen, Ⅱ, Math. Naturwiss. Anz. Ungar. Akad. Wiss, 24 (1906), 369–390.
[13]	S. Mehmood, F. Zafar, N. Asmin, New Hermite-Hadamard-Fejér type inequalities for $(\eta_{1}, \eta_{2})$-convex functions via fractional calculus, ScienceAsia, 46 (2020), 102–108. https://doi.org/10.2306/scienceasia1513-1874.2020.012 doi: 10.2306/scienceasia1513-1874.2020.012
[14]	S. M. Aslani, M. R. Delavar, S. M. Vaezpour, Inequalities of Fejér type related to generalized convex functions, Int. J. Anal. Appl., 16 (2018), 38–49.
[15]	M. Rostamian Delavar, S. Mohammadi Aslani, De La Sen, M. Hermite-Hadamard-Fejér inequality related to generalized convex functions via fractional integrals, J. Math., 2018 (2018). https://doi.org/10.1155/2018/5864091 doi: 10.1155/2018/5864091
[16]	M. E. Gordji, M. R. Delavar, M. De La Sen, On $\phi$-convex functions. J. Math. Inequalities, 10 (2016), 173–183. https://doi.org/10.7153/jmi-10-15 doi: 10.7153/jmi-10-15
[17]	M. E. Gordji, M. R. Delavar, S. S. Dragomir, Some inequalities related to $\eta$-convex functions, Preprint Rgmia Res. Rep. Coll., 18 (2015), 1–14.
[18]	M. R. Delavar, S. S. Dragomir, On $\eta$-convexity. J. Inequalities Appl., 20 (2017), 203–216. https://doi.org/10.7153/mia-20-14 doi: 10.7153/mia-20-14
[19]	M. Eshaghi, S. S. Dragomir, M. Rostamian Delavar, An inequality related to $\eta $-convex functions (Ⅱ), Int. J. Nonlinear Anal. Appl., 6 (2015), 27–33.
[20]	V. Jeyakumar, Strong and weak invexity in mathematical programming, University of Melbourne, Department of Mathematics, 55 (1984), 109–125.
[21]	A. Ben-Israel, B. Mond, What is invexity? J. Aust. Math. Soc., 28 (1986), 1–9. https://doi.org/10.1017/S0334270000005142 doi: 10.1017/S0334270000005142
[22]	M. A. Hanson, B. Mond, Convex transformable programming problems and invexity, J. Inform. Optim. Sci., 8 (1987), 201–207. https://doi.org/10.1080/02522667.1987.10698886 doi: 10.1080/02522667.1987.10698886
[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. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
[24]	S. S. Dragomir, Two mappings in connection to Hadamard's inequalities, J. Math. Anal. Appl., 167 (1992), 49–56. https://doi.org/10.1016/0022-247X(92)90233-4 doi: 10.1016/0022-247X(92)90233-4
[25]	A. Almutairi, A. Kilicman, New refinements of the Hadamard inequality on coordinated convex function, J. Inequal. Appl., 2019 (2019), 1–9.
[26]	S. S. Dragomir, Lebesgue integral inequalities of Jensen type for $\lambda $-convex functions, Armenian J. Math., 10 (2018), 1–19. https://doi.org/10.1186/s13660-019-2143-2 doi: 10.1186/s13660-019-2143-2
[27]	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. https://doi.org/10.1016/S0893-9659(98)00086-X doi: 10.1016/S0893-9659(98)00086-X
[28]	M. Samraiz, F. Nawaz, S. Iqbal, T. Abdeljawad, G. Rahman, K. S. Nisar, Certain mean-type fractional integral inequalities via different convexities with applications, J. Inequal. Appl., 2020 (2020), 1–19. https://doi.org/10.1186/s13660-020-02474-x doi: 10.1186/s13660-020-02474-x
[29]	C. Niculescu, L. E. Persson, Convex functions and their applications, (pp. xvi+-255), New York: Springer.
[30]	B. G. Pachpatte, On some integral inequalities involving convex functions, RGMIA Res. Rep. Collect., 3 (2000).
[31]	M. Tunc, On some new inequalities for convex functions, Turkish J. Math., 36 (2012), 245–251.
[32]	O. Almutairi, A. Kilicman, New fractional inequalities of midpoint type via s-convexity and their application, J. Inequal. Appl., 2019 (2019), 1–19. https://doi.org/10.1186/s13660-019-2215-3 doi: 10.1186/s13660-019-2215-3
[33]	O. Alabdali, A. Guessab, G. Schmeisser, Characterizations of uniform convexity for differentiable functions. Appl. Anal. Discrete Math., 13 (2019), 721–732. https://doi.org/10.2298/AADM190322029A doi: 10.2298/AADM190322029A
[34]	A. Guessab, O. Nouisser, G. Schmeisser, Enhancement of the algebraic precision of a linear operator and consequences under positivity, Positivity, 13 (2009), 693–707. https://doi.org/10.1007/s11117-008-2253-4 doi: 10.1007/s11117-008-2253-4
[35]	A. Guessab, Generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes, Comput. Math. Appl., 66 (2013), 1120–1136. https://doi.org/10.1016/j.camwa.2013.07.014 doi: 10.1016/j.camwa.2013.07.014
[36]	J. E. Peajcariaac, Y. L. Tong, Convex functions, partial orderings, and statistical applications, (1992), Academic Press. https://doi.org/10.1016/S0076-5392(08)62813-1
[37]	X. Qiang, G. Farid, M. Yussouf, K. A. Khan, A. U. Rahman, New generalized fractional versions of Hadamard and Fejér inequalities for harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1–13. https://doi.org/10.1186/s13660-020-02457-y doi: 10.1186/s13660-020-02457-y
[38]	I. Iscan, S. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237–244. https://doi.org/10.1016/j.amc.2014.04.020 doi: 10.1016/j.amc.2014.04.020
[39]	D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Ann. Univ. Craiova-Mat., 34 (2007), 82–87.
[40]	S. S. Dragomir, C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Mathematics Preprint Archive, 2003 (2003), 463–817.
[41]	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. https://doi.org/10.1016/j.jmaa.2016.09.018 doi: 10.1016/j.jmaa.2016.09.018
[42]	S. S. Dragomir, Integral inequalities of Jensen type for$\lambda$ -convex functions, Mat. Vestn., 68 (2016), 45–57.
[43]	M. E. Ozdemir, Some inequalities for the s-Godunova-Levin type functions, Math. Sci., 9 (2015), 27–32. https://doi.org/10.1007/s40096-015-0144-y doi: 10.1007/s40096-015-0144-y
[44]	S. Varosanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), 303–311.
[45]	O. Almutairi, A. Kilicman, Some integral inequalities for h-Godunova-Levin preinvexity. Symmetry, 11 (2019), 1500. https://doi.org/10.3390/sym11121500 doi: 10.3390/sym11121500
[46]	G. H. Toader, Some generalizations of the convexity, In: Proc. Colloq. Approx. Optim, Cluj Napoca (Romania), 1984,329–338.
[47]	M. Rostamian Delavar, S. Mohammadi Aslani, M. De La Sen, Hermite-Hadamard-Fejér inequality related to generalized convex functions via fractional integrals, J. Math., 2018 (2018). https://doi.org/10.1155/2018/5864091 doi: 10.1155/2018/5864091
[48]	E. K. Godunova, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, Numer. Math. Math. Phys., 138 (1985), 166.
[49]	E. D. Rainville, Special Functions, Chelsea Publ. Co., Bronx, 1971, New York.
[50]	R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, arXiv preprint, (2008). arXiv: 0805.3823.
[51]	S. S. Dragomir, S. Fitzpatrick, The Hadamard inequalities for s-convex functions in the second sense, Demonstr. Math., 32 (1999), 687–696. https://doi.org/10.1515/dema-1999-0403 doi: 10.1515/dema-1999-0403
[52]	I. Iscan, Hermite-Hadamard's inequalities for preinvex functions via fractional integrals and related fractional inequalities, arXiv Preprint, (2012). arXiv: 1204.0272.
[53]	M. Muddassar, M. I. Bhatti, M. Iqbal, Some new s-Hermite-Hadamard type inequalities for differentiable functions and their applications, Proc. Pakistan Academy Sci., 49 (2012), 9–17.
[54]	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. https://doi.org/10.1016/S0893-9659(98)00086-X doi: 10.1016/S0893-9659(98)00086-X
[55]	M. A. Noor, K. I. Noor, M. U. Awan, S. Khan, Hermite-Hadamard inequalities for s-Godunova-Levin preinvex functions, J. Adv. Math. Studies, 7 (2014), 12–19.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)