Research article

Generalized inequalities for integral operators via several kinds of convex functions

  • Received: 17 March 2020 Accepted: 09 May 2020 Published: 25 May 2020
  • MSC : 26A51, 26A33, 26D15

  • This paper investigates the bounds of an integral operator for several kinds of convex functions. By applying definition of (h - m)-convex function upper bounds of left sided (1.12) and right sided (1.13) integral operators are formulated which particularly provide upper bounds of various known conformable and fractional integrals. Further a modulus inequality is investigated for differentiable functions whose derivative in absolute value are (h - m)-convex. Moreover a generalized Hadamard inequality for (h - m)-convex functions is proved by utilizing these operators. Also all the results are obtained for (α, m)-convex functions. Finally some applications of proved results are discussed.

    Citation: Yue Wang, Ghulam Farid, Babar Khan Bangash, Weiwei Wang. Generalized inequalities for integral operators via several kinds of convex functions[J]. AIMS Mathematics, 2020, 5(5): 4624-4643. doi: 10.3934/math.2020297

    Related Papers:

  • This paper investigates the bounds of an integral operator for several kinds of convex functions. By applying definition of (h - m)-convex function upper bounds of left sided (1.12) and right sided (1.13) integral operators are formulated which particularly provide upper bounds of various known conformable and fractional integrals. Further a modulus inequality is investigated for differentiable functions whose derivative in absolute value are (h - m)-convex. Moreover a generalized Hadamard inequality for (h - m)-convex functions is proved by utilizing these operators. Also all the results are obtained for (α, m)-convex functions. Finally some applications of proved results are discussed.


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    [1] C. P. Niculescu, L. E. Persson, Convex functions and their applications. A contemporary approach, Springer Science+Business Media, Inc., 2006.
    [2] J. E. Pečarić, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications. Academics Press, New York, 1992.
    [3] A. W. Roberts, D. E. Varberg, Convex Functions; Academic Press: New York, NY, USA, 1973.
    [4] M. E. Özdemir, A. O. Akdemir, E. Set, On (h - m)-convexity and Hadamard-type inequalities, Transylv. J. Math. Mech., 8 (2016), 51-58.
    [5] V. G. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex, Cluj-Napoca, Romania, 1993.
    [6] A. U. Rehman, G. Farid, Q. U. Ain, Hermite-Hadamard type inequalities for (h - m)-convexity, Electron. J. Math. Anal. Appl., 6 (2018), 317-329.
    [7] S. M. Kang, G. Farid, W. Nazeer, et al. (h - m)-convex functions and associated fractional Hadamard and Fejér-Hadamard inequalities via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 78.
    [8] M. K. Bakula, M. E. Özdemir, J. Pečarić, Hadamard-type inequalities for m-convex and (α, m)- convex functions, J. Inequal. Pure. Appl. Math., 9 (2007), Article 96.
    [9] H. K. İşcan, M. Kadakal, Some new integral inequalities for functions whose nth derivatives in absolute value are (α, m)-convex functions, New Trends Math. Sci., 5 (2017), 180-185.
    [10] S. M. Kang, G. Farid, M. Waseem, et al. Generalized k-fractional integral inequalities associated with (α, m)-convex functions, J. Inequal. Appl., 2019 (2019), 255.
    [11] E. Set, M. Sardari, M. E. Özdemir, et al. On generalizations of the Hadamard inequality for (α, m)- convex functions, RGMIA Res. Rep. Coll., 12 (2009), Article 4.
    [12] W. Sun, Q. Liu, New Hermite-Hadamard type inequalities for (α, m)-convex functions and applications to special means, J. Math. Ineq., 11 (2017), 383-397.
    [13] A. O. Akdemir, E. Ekinci, E. Set, Conformable fractional integrals and related new integral inequalities, J. Nonlinear Comp. Anal., 18 (2017), 661-674.
    [14] S. Rashid, M. A. Noor, K. I. Noor, Fractional exponentially m-convex functions and inequalities, Inter. J. Inequal., 2019.
    [15] S. Rashid, M. A. Noor, K. I. Noor, New estimates exponentially convex functions via conformable fractional operator, Fractal and Fractional, 2019.
    [16] S. Mubeen, G. M. Habibullah, k-fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), 89-94.
    [17] A. A. Kilbas, H. M. Srivastava, J. J Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006.
    [18] Y. C. Kwun, G. Farid, W. Nazeer, et al. Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities, IEEE Access, 6 (2018), 64946-64953. doi: 10.1109/ACCESS.2018.2878266
    [19] G. Farid, Existence of an integral operator and its consequences in fractional and conformable integrals, Open J. Math. Sci., 3 (2019), 210-216. doi: 10.30538/oms2019.0064
    [20] 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. doi: 10.1016/j.jmaa.2016.09.018
    [21] S. S. Dragomir, Inequalities of Jensen's type for generalized k - g-fractional integrals of functions for which the composite fog-1 is convex, RGMIA Res. Rep. Coll. 20 (2017), Art. 133, 24.
    [22] S. Habib, S. Mubeen, M. N. Naeem, Chebyshev type integral inequalities for generalized kfractional conformable integrals, J. Inequal. Spec. Funct., 9 (2018), 53-65.
    [23] F. Jarad, E. Ugurlu, T. Abdeljawad, et al. On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), 247.
    [24] T. U. Khan, M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 316 (2019), 378-389.
    [25] A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional integrals and derivatives. Theory and Applications, Gordon and Breach, Switzerland, 1993.
    [26] M. Z. Sarıkaya, M. Dahmani, M. E. Kiriş, et al. (k, s)-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45 (2016), 77-89.
    [27] T. Tunc, H. Budak, F. Usta, et al. On new generalized fractional integral operators and related fractional inequalities, Available from: https://www.researchgate.net/publication/313650587.
    [28] G. Farid, Some Riemann-Liouville fractional integral for inequalities for convex functions, J. Anal., 2018. Available from: doi.org/10.1007/s41478-0079-4.
    [29] K. S. Nisar, G. Rahman, A. Khan, Some new inequalities for generalized fractional conformable integral operators, Adv. Difference Equ., 2019 (2019), 247.
    [30] K. S. Nisar, G. Rahman, K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 2019 (2019), 245.
    [31] K. S. Nisar, A. Tassaddiq, G. Rahman, et al. Some inequalities via fractional conformable integral operators, J. Inequal. Appl., 2019 (2019), 217.
    [32] G. Rahman, Z. Ullah, A. Khan, et al. Certain Chebyshev-Type inequalities involving fractional conformable integral operators, Mathematics, 7 (2019), 364.
    [33] G. Farid, W. Nazeer, M. S. Saleem, et al. Bounds of Riemann-Liouville fractional integrals in general form via convex functions and their Applications, Mathematics, 2018 (2018), 248.
    [34] L. N. Mishra, G. Farid, B. K. Bangash, Bounds of an integral operator for convex functions and results in fractional calculus, Honam Math. J., Appear.
    [35] G. Farid, Bounds of Riemann-Liouville fractional integral operators, Comput. Methods Differ. Equ., Appear.
    [36] M. Z. Sarıkaya, F. Ertuğral, On the generalized Hermite-Hadamard inequalities, Available from: https://www.researchgate.net/publication/321760443.
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