On Hadamard inequalities for refined convex functions via strictly monotone functions

Moquddsa Zahra; Dina Abuzaid; Ghulam Farid; Kamsing Nonlaopon; Moquddsa Zahra; Dina Abuzaid; Ghulam Farid; Kamsing Nonlaopon

doi:10.3934/math.20221096

AIMS Mathematics

2022, Volume 7, Issue 11: 20043-20057. doi: 10.3934/math.20221096

Previous Article Next Article

Research article

On Hadamard inequalities for refined convex functions via strictly monotone functions

1.
Department of Mathematics, University of Wah, Wah Cantt, Pakistan
2.
Department of Mathematics, King Abdul Aziz University, Saudi Arabia
3.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
4.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 23 July 2022 Revised: 22 August 2022 Accepted: 31 August 2022 Published: 13 September 2022
MSC : 26D10, 31A10, 26A33

In this paper, we define refined $ (\alpha, h-m) $-convex function with respect to a strictly monotone function. This function provides refinements of various well-known classes of functions for specific strictly monotone functions. By applying definition of this new function we prove the Hadamard inequalities for Riemann-Liouville fractional integrals. These inequalities give the refinements of fractional Hadamard inequalities for convex, $ (\alpha, m) $-convex, $ (h-m) $-convex, $ (s, m) $-convex, $ h $-convex and many other related well-known classes of functions implicitly. Also, Hadamard type inequalities for $ k $-fractional integrals are given.
- convex function,
- refined $ (\alpha, h-m) $-convex function,
- monotone function,
- Hadamard inequality,
- Riemann-Liouville fractional integrals
Citation: Moquddsa Zahra, Dina Abuzaid, Ghulam Farid, Kamsing Nonlaopon. On Hadamard inequalities for refined convex functions via strictly monotone functions[J]. AIMS Mathematics, 2022, 7(11): 20043-20057. doi: 10.3934/math.20221096

Related Papers:

Abstract

In this paper, we define refined $ (\alpha, h-m) $-convex function with respect to a strictly monotone function. This function provides refinements of various well-known classes of functions for specific strictly monotone functions. By applying definition of this new function we prove the Hadamard inequalities for Riemann-Liouville fractional integrals. These inequalities give the refinements of fractional Hadamard inequalities for convex, $ (\alpha, m) $-convex, $ (h-m) $-convex, $ (s, m) $-convex, $ h $-convex and many other related well-known classes of functions implicitly. Also, Hadamard type inequalities for $ k $-fractional integrals are given.

References

[1]	G. A. Anastassiou, Generalized fractional Hermite-Hadamard inequalities involving $m$-convexity and $(s, m)$-convexity, Facta Univ. Ser. Math. Inform., 28 (2013), 107–126.
[2]	M. K. Bakula, M. E. Özdemir, J. Pečarić, Hadamard type inequalities for $m$-convex and $(\alpha, m)$-convex functions, J. Inequal. Pure Appl. Math., 9 (2008), 1–25.
[3]	S. S. Dragomir, J. Pečariç, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335–341.
[4]	G. Farid, A. U. Rehman, Q. U. Ain, $k$-fractional integral inequalities of Hadamard type for $(h-m)$-convex functions, Comput. Methods Differ. Equ., 8 (2020), 119–140. https://doi.org/10.22034/CMDE.2019.9462 doi: 10.22034/CMDE.2019.9462
[5]	E. Set, B. Çelik, Fractional Hermite-Hadamard type inequalities for quasi-convex functions, Ordu Univ. J. Sci. Tech., 6 (2016), 137–149.
[6]	A. W. Roberts, D. E. Varberg, Convex functions, New York: Academic Press, 1973.
[7]	J. E. Pečarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings, and statistical applications, Academic Press, 1992.
[8]	M. Bombardelli, S. Varošanec, Properties of $h$-convex functions related to the Hermite-Hadamard-Fejér inequalities, Comput. Math. Appl., 58 (2009), 1869–1877. https://doi.org/10.1016/j.camwa.2009.07.073 doi: 10.1016/j.camwa.2009.07.073
[9]	H. Hudzik, L. Maligranda, Some remarks on $s$-convex functions, Aequationes Math., 48 (1994), 100–111. https://doi.org/10.1007/BF01837981 doi: 10.1007/BF01837981
[10]	V. G. Mihesan, A generalization of the convexity, In: Seminar on functional equations, approximation and convexity, Romania: Cluj-Napoca, 1993.
[11]	S. Mehmood, G. Farid, Fractional integral inequalities for exponentially $m$-convex functions, Open J. Math. Sci., 4 (2020), 78–85. https://doi.org/10.30538/oms2020.0097 doi: 10.30538/oms2020.0097
[12]	M. E. Özdemir, 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
[13]	T. Yan, G. Farid, H. Yasmeen, C. Y. Jung, On Hadamard type fractional inequalities for Riemann-Liouville integrals via a generalized convexity, Fractal Fract., 6 (2022), 1–15. https://doi.org/10.3390/fractalfract6010028 doi: 10.3390/fractalfract6010028
[14]	M. Zahra, M. Ashraf, G. Farid, K. Nonlaopon, Some new kinds of fractional integral inequalities via refined $(\alpha, h-m)$-convex function, Math. Probl. Eng., 2021 (2021), 1–15. https://doi.org/10.1155/2021/8331092 doi: 10.1155/2021/8331092
[15]	G. H. Toader, Some generalization of convexity, Proc. Colloq. Approx. Optim., 1984,329–338.
[16]	S. M. Yuan, Z. M. Liu, Some properties of $\alpha$-convex and $\alpha$-quasiconvex functions with respect to $n$-symmetric points, Appl. Math. Comput., 188 (2007), 1142–1150. https://doi.org/10.1016/j.amc.2006.10.060 doi: 10.1016/j.amc.2006.10.060
[17]	S. Hussain, M. I. Bhatti, M. Iqbal, Hadamard-type inequalities for $s$-convex functions I, Punjab Univ. J. Math., 41 (2009), 51–60.
[18]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[19]	S. Mubeen, G. M. Habibullah, $k$-fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), 89–94.
[20]	M. E. Özdemir, M. Avcı, E. Set, On some inequalities of Hermite-Hadamard type via $m$-convexity, Appl. Math. Lett., 23 (2010), 1065–1070. https://doi.org/10.1016/j.aml.2010.04.037 doi: 10.1016/j.aml.2010.04.037
[21]	M. A. Latif, S. S. Dragomir, On Hermite-Hadamard type integral inequalities for $n$-times differentiable log-preinvex functions, Filomat, 29 (2015), 1651–1661. https://doi.org/10.2298/FIL1507651L doi: 10.2298/FIL1507651L
[22]	W. Sudsutad, S. K. Ntouyas, J. Tariboon, Fractional integral inequalities via Hadamard's fractional integral, Abstr. Appl. Anal., 2014 (2014), 1–11. https://doi.org/10.1155/2014/563096 doi: 10.1155/2014/563096
[23]	M. E. Özdemir, M. Avcı-Ardıç, H. Kavurmaci-Önalan, Hermite-Hadamard type inequalities for $s$-convex and $s$-concave functions via fractional integrals, Turkish J. Sci., 1 (2016), 28–40.
[24]	M. Tunç, E. Göv, Ü. Şanal, On $tgs$-convex function and their inequalities, Facta Univ. Ser. Math. Inform., 30 (2015), 679–691.
[25]	G. Farid, M. Zahra, Y. C. Kwun, S. M. Kang, Fractional Hadamard-type inequalities for refined $(\alpha, h-m)-p$-convex function and their consequences, Math. Methods Appl. Sci., 2022, In press.

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)