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Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

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  • Waqar Afzal

    (Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan)

  • Mujahid Abbas

    (Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
    Department of Mechanical Engineering Sciences, Faculty of Engineering and the Built Environment, Doornfontein Campus, University of Johannesburg, Johannesburg 2092, South Africa
    Department of Medical Research, China Medical University, Taichung 406040, Taiwan)

  • Omar Mutab Alsalami

    (Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia)

Abstract

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space ( l q ( · ) ( L p ( · ) ) ). Moreover, it was developed using classical Lebesgue space ( L p ) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Suggested Citation

  • Waqar Afzal & Mujahid Abbas & Omar Mutab Alsalami, 2024. "Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces," Mathematics, MDPI, vol. 12(16), pages 1-33, August.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:16:p:2464-:d:1453375
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    References listed on IDEAS

    as
    1. Yong Wang & Zheng-Hai Huang & Liqun Qi, 2018. "Global Uniqueness and Solvability of Tensor Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 137-152, April.
    2. Hüseyin Budak & Fatih Hezenci & Hasan Kara & Mehmet Zeki Sarikaya, 2023. "Bounds for the Error in Approximating a Fractional Integral by Simpson’s Rule," Mathematics, MDPI, vol. 11(10), pages 1-16, May.
    3. Tong-tong Shang & Guo-ji Tang, 2023. "Mixed polynomial variational inequalities," Journal of Global Optimization, Springer, vol. 86(4), pages 953-988, August.
    4. Waqar Afzal & Khurram Shabbir & Mubashar Arshad & Joshua Kiddy K. Asamoah & Ahmed M. Galal & Ching-Feng Wen, 2023. "Some Novel Estimates of Integral Inequalities for a Generalized Class of Harmonical Convex Mappings by Means of Center-Radius Order Relation," Journal of Mathematics, Hindawi, vol. 2023, pages 1-14, June.
    5. Aqeel Ahmad Mughal & Hassan Almusawa & Absar Ul Haq & Imran Abbas Baloch & Ahmet Ocak Akdemir, 2021. "Properties and Bounds of Jensen-Type Functionals via Harmonic Convex Functions," Journal of Mathematics, Hindawi, vol. 2021, pages 1-13, August.
    6. Yahya Almalki & Waqar Afzal, 2023. "Some New Estimates of Hermite–Hadamard Inequalities for Harmonical cr - h -Convex Functions via Generalized Fractional Integral Operator on Set-Valued Mappings," Mathematics, MDPI, vol. 11(19), pages 1-21, September.
    7. Francesca Anceschi & Annamaria Barbagallo & Serena Guarino Lo Bianco, 2023. "Inverse Tensor Variational Inequalities and Applications," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 570-589, February.
    8. Xiaoju Zhang & Khurram Shabbir & Waqar Afzal & He Xiao & Dong Lin & Xiaolong Qin, 2022. "Hermite–Hadamard and Jensen-Type Inequalities via Riemann Integral Operator for a Generalized Class of Godunova–Levin Functions," Journal of Mathematics, Hindawi, vol. 2022, pages 1-12, August.
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