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Integral Inequalities for Some New Classes of Convex Functions

Received: 19 March 2015     Accepted: 21 March 2015     Published: 9 June 2015
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Abstract

In this paper, we introduce a new class of convex functions, which is called nonconvex functions. We show that this class unifies several previously known and new classes of convex functions. We derive several new inequalities of Hermite-Hadamard type for nonconvex functions. Some special cases are also discussed. Results proved in this paper continue to hold for these special cases.

Published in American Journal of Applied Mathematics (Volume 3, Issue 3-1)

This article belongs to the Special Issue Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015)

DOI 10.11648/j.ajam.s.2015030301.11
Page(s) 1-5
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Convex, Hermite-Hadamard’s Inequalities, Convex Functions

References
[1] W W. Breckner, Stetigkeitsaussagen fiir eine Klasse verallgemeinerter convexer funktionen in topologischen linearen Raumen. Pupl. Inst. Math. 23 (1978), 13-20.
[2] G.Cristescu and L. Lupsa, Non-connected Convexities and Applications. Kluwer Academic Publishers, Dordrecht, Holland, 2002.
[3] 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(2), (2015).
[4] S. S. Dragomir, Inequalities of Hermite-Hadamard type for -convex functions on linear spaces, preprint, (2014).
[5] S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications. Victoria University, Australia, 2000.
[6] S. S. Dragomir, S. Fitzpatrik, The Hadamard’s inequality for s-convex functions in the second sense, Demonstration Math., 32 (4) (1999), 687–696.
[7] S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21 (1995), 335-341.
[8] E. K. Godunova, V. I. Levin, Neravenstva dlja funkcii sirokogo klassa, soderzascego vypuklye, monotonnye i nekotorye drugie vidy funkii. Vycislitel. Mat. i. Fiz. Mezvuzov. Sb. Nauc. Trudov, MGPI, Moskva. (1985) 138-142, (in Russian).
[9] M. V. Mihai, M. A. Noor, K. I. Noor, M. U. Awan, Some integral inequalities for harmonic h-convex functions involving hypergeometric functions, Appl. Math. Comput. 252 (2015) 257–262.
[10] M.A. Noor, K.I. Noor, M.U. Awan, Generalized convexity and integral inequalities, Appl. Math. Inf. Sci. 9 (1) (2015), 233–243
[11] M.A. Noor, K.I. Noor, M.U. Awan, Integral inequalities for coordinated Harmonically convex functions, Complex Var. Elliptic Equ. (2014).
[12] M.A. Noor, K.I. Noor, M.U. Awan, S. Costache, Some integral inequalities for harmonically h-convex functions, U.P.B. Sci. Bull. Serai A. 77(1) 2015, 5-16.
[13] M.A. Noor, K.I. Noor, M.U. Awan, S. Khan, Fractional Hermite–Hadamard inequalities for some new classes of Godunova–Levin functions, Appl. Math. Inf. Sci. 8 (6) (2014), 2865–2872.
[14] M. Z. Sarikaya, A. Saglam, H. Yildirim, On some Hadamard–type inequalities for –convex functions, J. Math. Inequal. 2, 3(2008), 335–341.
[15] M. Z. Sarikaya, E. Set, M. E. Ozdemir, On some new inequalities of Hadamard type involving -convex functions, Acta Math. Univ. Comenianae LXXIX, 2(2010), 265-272.
[16] S. Varošanec, On -convexity, J. Math. Anal. Appl. 326 (2007), 303-311.
Cite This Article
  • APA Style

    Muhammad Aslam Noor, Khalida Inayat Noor, Muhammad Uzair Awan. (2015). Integral Inequalities for Some New Classes of Convex Functions. American Journal of Applied Mathematics, 3(3-1), 1-5. https://doi.org/10.11648/j.ajam.s.2015030301.11

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    ACS Style

    Muhammad Aslam Noor; Khalida Inayat Noor; Muhammad Uzair Awan. Integral Inequalities for Some New Classes of Convex Functions. Am. J. Appl. Math. 2015, 3(3-1), 1-5. doi: 10.11648/j.ajam.s.2015030301.11

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    AMA Style

    Muhammad Aslam Noor, Khalida Inayat Noor, Muhammad Uzair Awan. Integral Inequalities for Some New Classes of Convex Functions. Am J Appl Math. 2015;3(3-1):1-5. doi: 10.11648/j.ajam.s.2015030301.11

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  • @article{10.11648/j.ajam.s.2015030301.11,
      author = {Muhammad Aslam Noor and Khalida Inayat Noor and Muhammad Uzair Awan},
      title = {Integral Inequalities for Some New Classes of Convex Functions},
      journal = {American Journal of Applied Mathematics},
      volume = {3},
      number = {3-1},
      pages = {1-5},
      doi = {10.11648/j.ajam.s.2015030301.11},
      url = {https://doi.org/10.11648/j.ajam.s.2015030301.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.s.2015030301.11},
      abstract = {In this paper, we introduce a new class of convex functions, which is called nonconvex functions. We show that this class unifies several previously known and new classes of convex functions. We derive several new inequalities of Hermite-Hadamard type for nonconvex functions. Some special cases are also discussed. Results proved in this paper continue to hold for these special cases.},
     year = {2015}
    }
    

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    T1  - Integral Inequalities for Some New Classes of Convex Functions
    AU  - Muhammad Aslam Noor
    AU  - Khalida Inayat Noor
    AU  - Muhammad Uzair Awan
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    DO  - 10.11648/j.ajam.s.2015030301.11
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    UR  - https://doi.org/10.11648/j.ajam.s.2015030301.11
    AB  - In this paper, we introduce a new class of convex functions, which is called nonconvex functions. We show that this class unifies several previously known and new classes of convex functions. We derive several new inequalities of Hermite-Hadamard type for nonconvex functions. Some special cases are also discussed. Results proved in this paper continue to hold for these special cases.
    VL  - 3
    IS  - 3-1
    ER  - 

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Author Information
  • Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

  • Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

  • Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

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