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Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus

Author

Listed:
  • Surang Sitho

    (Department of Social and Applied Science, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    These authors contributed equally to this work.)

  • Muhammad Aamir Ali

    (Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
    These authors contributed equally to this work.)

  • Hüseyin Budak

    (Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, Turkey
    These authors contributed equally to this work.)

  • Sotiris K. Ntouyas

    (Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
    Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
    These authors contributed equally to this work.)

  • Jessada Tariboon

    (Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    These authors contributed equally to this work.)

Abstract

In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the q ϰ 2 -quantum integral to show midpoint and trapezoidal inequalities for q ϰ 2 -differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via q ϰ 1 -quantum integral is not valid for preinvex functions, and we present its proper form. We use q ϰ 1 -quantum integrals to show midpoint inequalities for q ϰ 1 -differentiable preinvex functions. It is also demonstrated that by considering the limit q → 1 − and η ϰ 2 , ϰ 1 = − η ϰ 1 , ϰ 2 = ϰ 2 − ϰ 1 in the newly derived results, the newly proved findings can be turned into certain known results.

Suggested Citation

  • Surang Sitho & Muhammad Aamir Ali & Hüseyin Budak & Sotiris K. Ntouyas & Jessada Tariboon, 2021. "Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus," Mathematics, MDPI, vol. 9(14), pages 1-21, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1666-:d:594910
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    References listed on IDEAS

    as
    1. Sikander Mehmood & Fiza Zafar & Nusrat Yasmin, 2019. "Hermite-Hadamard-Fejér Type Inequalities for Preinvex Functions Using Fractional Integrals," Mathematics, MDPI, vol. 7(5), pages 1-11, May.
    2. Noor, Muhammad Aslam & Noor, Khalida Inayat & Awan, Muhammad Uzair, 2015. "Some quantum integral inequalities via preinvex functions," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 242-251.
    3. Noor, Muhammad Aslam & Noor, Khalida Inayat & Awan, Muhammad Uzair, 2015. "Some quantum estimates for Hermite–Hadamard inequalities," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 675-679.
    4. Seksan Jhanthanam & Jessada Tariboon & Sotiris K. Ntouyas & Kamsing Nonlaopon, 2019. "On q -Hermite-Hadamard Inequalities for Differentiable Convex Functions," Mathematics, MDPI, vol. 7(7), pages 1-9, July.
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