Some weighted integral inequalities for differentiable h-preinvex functions

Muhammad Amer Latif 1 , Sever Silvestru Dragomir 2 , and Ebrahim Momoniat 3
  • 1 School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
  • 2 College of Engineering and Science, Victoria University, Mathematics, Melbourne City, Australia
  • 3 School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
Muhammad Amer Latif
  • Corresponding author
  • School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Sever Silvestru Dragomir
  • Mathematics, College of Engineering and Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia, and School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Ebrahim Momoniat
  • School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

In this paper, by using a weighted identity for functions defined on an open invex subset of the set of real numbers, by using the Hölder integral inequality and by using the notion of h-preinvexity, we present weighted integral inequalities of Hermite–Hadamard-type for functions whose derivatives in absolute value raised to certain powers are h-preinvex functions. Some new Hermite–Hadamard-type integral inequalities are obtained when h is super-additive. Inequalities of Hermite–Hadamard-type for s-preinvex functions are given as well as a special case of our results.

  • [1]

    H. Alzer, A superadditive property of Hadamard’s gamma function, Abh. Math. Semin. Univ. Hambg. 79 (2009), no. 1, 11–23.

    • Crossref
    • Export Citation
  • [2]

    T. Antczak, Mean value in invexity analysis, Nonlinear Anal. 60 (2005), no. 8, 1473–1484.

    • Crossref
    • Export Citation
  • [3]

    A. Barani, A. G. Ghazanfari and S. S. Dragomir, Hermite–Hadamard inequality for functions whose derivatives absolute values are preinvex, J. Inequal. Appl. 2012 (2012), Paper No. 247.

  • [4]

    S. S. Dragomir, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl. 167 (1992), no. 1, 49–56.

    • Crossref
    • Export Citation
  • [5]

    S. S. Dragomir and 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), no. 5, 91–95.

    • Crossref
    • Export Citation
  • [6]

    D.-Y. Hwang, Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables, Appl. Math. Comput. 217 (2011), no. 23, 9598–9605.

  • [7]

    D.-Y. Hwang, Some inequalities for differentiable convex mapping with application to weighted midpoint formula and higher moments of random variables, Appl. Math. Comput. 232 (2014), 68–75.

  • [8]

    İ. İscan, Hermite–Hadamard’s inequalities for preinvex function via fractional integrals and related fractional inequalities, Amer. J. Math. Anal. 1 (2013), no. 3, 33–38.

  • [9]

    U. S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 147 (2004), no. 1, 137–146.

  • [10]

    U. S. Kırmacı and M. E. Özdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 153 (2004), no. 2, 361–368.

  • [11]

    M. A. Latif, On Hermite–Hadamard type integral inequalities for n-times differentiable preinvex functions with applications, Stud. Univ. Babeş-Bolyai Math. 58 (2013), no. 3, 325–343.

  • [12]

    M. A. Latif, Inequalities of Hermite–Hadamard type for functions whose derivatives in absolute value are convex with applications, Arab J. Math. Sci. 21 (2015), no. 1, 84–97.

  • [13]

    M. A. Latif and S. S. Dragomir, New inequalities of Hermite–Hadamard type for functions whose derivatives in absolute value are convex with applications, Acta Univ. M. Belii Ser. Math. 2013 (2013), 24–39.

  • [14]

    M. A. Latif and S. S. Dragomir, Some Hermite–Hadamard type inequalities for functions whose partial derivatives in absolute value are preinvex on the co-ordinates, Facta Univ. Ser. Math. Inform. 28 (2013), no. 3, 257–270.

  • [15]

    M. A. Latif and S. S. Dragomir, Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications, J. Inequal. Appl. 2013 (2013), Paper No. 575.

  • [16]

    M. A. Latif and S. S. Dragomir, Generalization of Hermite–Hadamard type inequalities for n-times differentiable functions which are s-preinvex in the second sense with applications, Hacet. J. Math. Stat. 44 (2015), no. 4, 839–853.

  • [17]

    M. A. Latif, S. S. Dragomir and E. Momoniat, Some weighted Hermite–Hadamard–Noor type inequalities for differentiable preinvex and quasi preinvex functions, Punjab Univ. J. Math. (Lahore) 47 (2015), no. 1, 57–72.

  • [18]

    A. Lupaş, A generalization of Hadamard inequalities for convex functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544–576 (1976), 115–121.

  • [19]

    M. Matloka, On some Hadamard-type inequalities for ( h 1 , h 2 ) {(h_{1},h_{2})}-preinvex functions on the co-ordinates, J. Inequal. Appl. 2013 (2013), Paper No. 227.

  • [20]

    M. Matloka, Inequalities for h-preinvex functions, Appl. Math. Comput. 234 (2014), 52–57.

  • [21]

    M. Matloka, On some new inequalities for differentiable (h 1 , h 2 {h_{1},h_{2}})-preinvex functions on the co-ordinates, Math. Stat. 2 (2014), no. 2, 6–14.

  • [22]

    M. A. Noor, Hermite–Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory 2 (2007), no. 2, 126–131.

  • [23]

    M. A. Noor, On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. Pure Appl. Math. 8 (2007), no. 3, Paper No. 75.

  • [24]

    C. E. M. Pearce and J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett. 13 (2000), no. 2, 51–55.

    • Crossref
    • Export Citation
  • [25]

    F. Qi, Z.-L. Wei and Q. Yang, Generalizations and refinements of Hermite–Hadamard’s inequality, Rocky Mountain J. Math. 35 (2005), no. 1, 235–251.

    • Crossref
    • Export Citation
  • [26]

    A. Saglam, H. Yıldırım and M. Z. Sarikaya, Some new inequalities of Hermite–Hadamard’s type, Kyungpook Math. J. 50 (2010), no. 3, 399–410.

    • Crossref
    • Export Citation
  • [27]

    M. Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Modelling 54 (2011), no. 9–10, 2175–2182.

    • Crossref
    • Export Citation
  • [28]

    M. Z. Sarikaya, M. Avci and H. Kavurmaci, On some inequalities of Hermite–Hadamard type for convex functions, AIP Conf. Proc. 1309 (2010), 10.1063/1.3525218.

  • [29]

    M. Z. Sarikaya, H. Bozkurt and N. Alp, On Hermite–Hadamard type integral inequalities for preinvex and log-preinvex functions, Contemp. Anal. Appl. Math. 1 (2013), no. 2, 237–252.

  • [30]

    S. Varošanec, On h-convexity, J. Math. Anal. Appl. 326 (2007), no. 1, 303–311.

    • Crossref
    • Export Citation
  • [31]

    S.-H. Wang and F. Qi, Hermite–Hadamard type inequalities for n-times differentiable and preinvex functions, J. Inequal. Appl. 2014 (2014), Paper No. 49.

  • [32]

    Y. Wang, B.-Y. Xi and F. Qi, Hermite–Hadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex, Matematiche (Catania) 69 (2014), no. 1, 89–96.

  • [33]

    T. Weir and B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl. 136 (1988), no. 1, 29–38.

    • Crossref
    • Export Citation
  • [34]

    S.-H. Wu, On the weighted generalization of the Hermite–Hadamard inequality and its applications, Rocky Mountain J. Math. 39 (2009), no. 5, 1741–1749.

    • Crossref
    • Export Citation
  • [35]

    G.-S. Yang, D.-Y. Hwang and K.-L. Tseng, Some inequalities for differentiable convex and concave mappings, Comput. Math. Appl. 47 (2004), no. 2–3, 207–216.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Search