Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus

Saad Ihsan Butt; Muhammad Nasim Aftab; Hossam A. Nabwey; Sina Etemad; Saad Ihsan Butt; Muhammad Nasim Aftab; Hossam A. Nabwey; Sina Etemad

doi:10.3934/math.2024268

AIMS Mathematics

2024, Volume 9, Issue 3: 5523-5549. doi: 10.3934/math.2024268

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Research article

Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
2.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
3.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
4.
Mathematics in Applied Sciences and Engineering Research Group, Scientific Research Center, Al-Ayen University, Nasiriyah 64001, Iraq

Received: 20 November 2023 Revised: 18 January 2024 Accepted: 22 January 2024 Published: 29 January 2024
MSC : 05A30, 26A51, 26D10, 26D15

The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval $ [\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $, we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at $ \mathrm{b_{0}}\in[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point $ \mathrm{b_{1}} $, and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus.
- Hermite-Hadamard inequality,
- convex functions,
- symmetric quantum calculus
Citation: Saad Ihsan Butt, Muhammad Nasim Aftab, Hossam A. Nabwey, Sina Etemad. Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus[J]. AIMS Mathematics, 2024, 9(3): 5523-5549. doi: 10.3934/math.2024268

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Abstract

The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval $ [\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $, we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at $ \mathrm{b_{0}}\in[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point $ \mathrm{b_{1}} $, and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus.

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