Research article

On $\mathscr{M}$-convex functions

  • Received: 30 December 2019 Accepted: 25 February 2020 Published: 04 March 2020
  • MSC : 05A30, 26A33, 26A51, 26D15

  • In this article, we introduce the notion of $\mathscr{M}$-convex functions, $\log$-$\mathscr{M}$-convex functions and the notion of quasi $\mathscr{M}$-convex functions. We derive some new analogues of Hermite-Hadamard like inequalities associated with $\mathscr{M}$-convex functions by using the concepts of ordinary, fractional and quantum calculus. The main results of this paper may be useful where bounds for natural phenomena described by integrals such as mechanical work are frequently required. These results are also helpful in the field of numerical analysis where error analysis is required.

    Citation: Muhammad Uzair Awan, Muhammad Aslam Noor, Tingsong Du, Khalida Inayat Noor. On $\mathscr{M}$-convex functions[J]. AIMS Mathematics, 2020, 5(3): 2376-2387. doi: 10.3934/math.2020157

    Related Papers:

  • In this article, we introduce the notion of $\mathscr{M}$-convex functions, $\log$-$\mathscr{M}$-convex functions and the notion of quasi $\mathscr{M}$-convex functions. We derive some new analogues of Hermite-Hadamard like inequalities associated with $\mathscr{M}$-convex functions by using the concepts of ordinary, fractional and quantum calculus. The main results of this paper may be useful where bounds for natural phenomena described by integrals such as mechanical work are frequently required. These results are also helpful in the field of numerical analysis where error analysis is required.


    加载中


    [1] M. Alp, M. Z. Sarikaya, M. Kunt, et al. q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, J. King Saud Univ., Sci., 30 (2018), 193-203. doi: 10.1016/j.jksus.2016.09.007
    [2] M. U. Awan, G. Cristescu, M. A. Noor, et al. Upper and lower bounds for Riemann type quantum integrals of preinvex and preinvex dominated functions, UPB Sci. Bull., Ser. A., 79 (2017), 33-44.
    [3] F. Al-Azemi, O. Calin, Asian options with harmonic average, Appl. Math. Inf. Sci., 9 (2015), 1-9.
    [4] G. Cristescu, M. A. Noor, M. U. Awan, Bounds of the second degree cumulative frontier gaps of functions with generalized convexity, Carpath. J. Math., 31 (2015), 173-180.
    [5] G. Cristescu, L. Lupsa, Non-connected Convexities and Applications, Kluwer Academic Publishers, Dordrecht, Holland, 2002.
    [6] 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.
    [7] S. S. Dragomir, B. Mond, Integral inequalities of Hadamard's type for log-convex functions, Demonstration Math., 2 (1998), 354-364.
    [8] S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Victoria University, Australia, 2000.
    [9] 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 (2016), 3112-3126. doi: 10.22436/jnsa.009.05.102
    [10] 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 (2017), 358-369.
    [11] A. Guessab, G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115 (2002), 260-288. doi: 10.1006/jath.2001.3658
    [12] A. Guessab, G. Schmeisser, Convexity results and sharp error estimates in approximate multivariate integration, Math. Comp., 73 (2004), 1365-1384.
    [13] I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe J. Math. Stat., 43 (2014), 935-942.
    [14] M. Khaled, P. Agarwal, New Hermite-Hadamard type integral inequalities for convex functions and their applications, J. Comput. Appl. Math., 350 (2019), 274-285. doi: 10.1016/j.cam.2018.10.022
    [15] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional dierential equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
    [16] K. Liu, J. R. Wang, D. O'Regan, On the Hermite-Hadamard type inequality for ψ-RiemannLiouville fractional integrals via convex functions, J. Inequal. Appl., 2019 (2019), 27.
    [17] M. V. Mihai, M. A. Noor, K. I. Noor, et al. Some integral inequalities for harmonic h-convex functions involving hypergeometric functions, Appl. Math. Comput., 252 (2015), 257-262.
    [18] C. P. Niculescu, L. E. Persson, Convex Functions and Their Applications. A Contemporary Approach, 2 Eds. CMS Books in Mathematics, vol. 23. Springer, New York, 2018.
    [19] M. A. Noor, K. I. Noor, M. U. Awan, Some quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput., 251 (2015), 675-679.
    [20] M. A. Noor, K. I. Noor, M. U. Awan, et al. Fractional Hermite-Hadamard inequalities for some new classes of Godunova-Levin functions, Appl. Math. Inf. Sci., 8 (2014), 2865-2872. doi: 10.12785/amis/080623
    [21] C. E. M. Pearce, J. Pecaric, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13 (2000), 51-55.
    [22] J. E. Pecaric, F. Prosch, Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, NewYork, 1992.
    [23] L. Riahi, M. U. Awan, M. A. Noor, Some complementary q-bounds via different classes of convex functions, UPB Sci. Bull., Ser. A, 79 (2017), 171-182.
    [24] M. Z. Sarikaya, E. Set, H. Yaldiz, et al. Hermite-Hadamards inequalities for fractional in- tegrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403-2407. doi: 10.1016/j.mcm.2011.12.048
    [25] J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 121.
    [26] Y. Zhang, T. S. Du, H. Wang, et al. Different types of quantum integral inequalities via (α, m)- convexity, J. Inequal. Appl., 2018 (2018), 264.
  • Reader Comments
  • © 2020 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2906) PDF downloads(373) Cited by(1)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog