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Some ( p , q )-Estimates of Hermite-Hadamard-Type Inequalities for Coordinated Convex and Quasi- Convex Functions

Author

Listed:
  • Humaira Kalsoom

    (School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
    Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan)

  • Muhammad Amer

    (Department of Basic Sciences, Deanship of Preparatory Year Program, University of Hail, Hail 2440, Saudi Arabia)

  • Moin-ud-Din Junjua

    (Department of Mathematics and Statistics, Institute of Southern Punjab, Multan 32100, Pakistan)

  • Sabir Hussain

    (Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan)

  • Gullnaz Shahzadi

    (Department of Mechanical Engineering, Ecole de Technologie Superieure, 1100 Notre-Dame W, Montreal, QC H3C 1K3, Canada)

Abstract

In this paper, we present the preliminaries of ( p , q ) -calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over [ a , b ] × [ c , d ] by using the ( p , q ) -calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the ( p , q ) -calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the ( p , q ) -calculus of the functions of two variables.

Suggested Citation

  • Humaira Kalsoom & Muhammad Amer & Moin-ud-Din Junjua & Sabir Hussain & Gullnaz Shahzadi, 2019. "Some ( p , q )-Estimates of Hermite-Hadamard-Type Inequalities for Coordinated Convex and Quasi- Convex Functions," Mathematics, MDPI, vol. 7(8), pages 1-22, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:683-:d:253406
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    References listed on IDEAS

    as
    1. 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.
    2. Hefeng Zhuang & Wenjun Liu & Jaekeun Park, 2019. "Some Quantum Estimates of Hermite-Hadamard Inequalities for Quasi-Convex Functions," Mathematics, MDPI, vol. 7(2), pages 1-18, February.
    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.
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    Cited by:

    1. Julalak Prabseang & Kamsing Nonlaopon & Jessada Tariboon & Sotiris K. Ntouyas, 2021. "Refinements of Hermite–Hadamard Inequalities for Continuous Convex Functions via ( p , q )-Calculus," Mathematics, MDPI, vol. 9(4), pages 1-12, February.
    2. Saima Rashid & Muhammad Aslam Noor & Khalida Inayat Noor & Farhat Safdar & Yu-Ming Chu, 2019. "Hermite-Hadamard Type Inequalities for the Class of Convex Functions on Time Scale," Mathematics, MDPI, vol. 7(10), pages 1-20, October.

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