A novel fractional approach to finding the upper bounds of Simpson and Hermite-Hadamard-type inequalities in tensorial Hilbert spaces by using differentiable convex mappings

Zareen A. Khan; Waqar Afzal; Mujahid Abbas; Jongsuk Ro; Najla M. Aloraini; Zareen A. Khan; Waqar Afzal; Mujahid Abbas; Jongsuk Ro; Najla M. Aloraini

doi:10.3934/math.20241671

AIMS Mathematics

2024, Volume 9, Issue 12: 35151-35180. doi: 10.3934/math.20241671

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A novel fractional approach to finding the upper bounds of Simpson and Hermite-Hadamard-type inequalities in tensorial Hilbert spaces by using differentiable convex mappings

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan
3.
Department of Mechanical Engineering Sciences, Faculty of Engineering and the Built Environment, Doornfontein Campus, University of Johannesburg, South Africa
4.
Department of Medical Research, China Medical University, Taichung 406040, Taiwan
5.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
6.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
7.
Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia

Received: 28 August 2024 Revised: 27 November 2024 Accepted: 04 December 2024 Published: 16 December 2024
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

Function spaces are significant in the study and application of mathematical inequalities. The objective of this article is to develop several new bounds and refinements for well-known inequalities that involve Hilbert spaces within a tensorial framework. Using self-adjoint operators in tensor Hilbert spaces, we developed Simpson type inequalities by using different types of generalized convex mappings. Our next step involved developing a variety of new variations of the Hermite and Hadamard inequalities using convex mappings with some special means, specifically arithmetic and geometric means. Furthermore, we developed a number of new fractional identities, which are used in our main findings, by using Riemann-Liouville integrals. In addition, we discuss some examples involving log convex functions and their consequences.
- Hermite-Hadamard,
- fractional calculus,
- upper bounds,
- mathematical operators
Citation: Zareen A. Khan, Waqar Afzal, Mujahid Abbas, Jongsuk Ro, Najla M. Aloraini. A novel fractional approach to finding the upper bounds of Simpson and Hermite-Hadamard-type inequalities in tensorial Hilbert spaces by using differentiable convex mappings[J]. AIMS Mathematics, 2024, 9(12): 35151-35180. doi: 10.3934/math.20241671

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Abstract

Function spaces are significant in the study and application of mathematical inequalities. The objective of this article is to develop several new bounds and refinements for well-known inequalities that involve Hilbert spaces within a tensorial framework. Using self-adjoint operators in tensor Hilbert spaces, we developed Simpson type inequalities by using different types of generalized convex mappings. Our next step involved developing a variety of new variations of the Hermite and Hadamard inequalities using convex mappings with some special means, specifically arithmetic and geometric means. Furthermore, we developed a number of new fractional identities, which are used in our main findings, by using Riemann-Liouville integrals. In addition, we discuss some examples involving log convex functions and their consequences.

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