Dynamic inequalities of Grüss, Ostrowski and Trapezoid type via diamond-$ \alpha $ integrals and Montgomery identity

Marwa M. Tharwat; Marwa M. Ahmed; Ammara Nosheen; Khuram Ali Khan; Iram Shahzadi; Dumitru Baleanu; Ahmed A. El-Deeb; Marwa M. Tharwat; Marwa M. Ahmed; Ammara Nosheen; Khuram Ali Khan; Iram Shahzadi; Dumitru Baleanu; Ahmed A. El-Deeb

doi:10.3934/math.2024624

AIMS Mathematics

2024, Volume 9, Issue 5: 12778-12799. doi: 10.3934/math.2024624

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Dynamic inequalities of Grüss, Ostrowski and Trapezoid type via diamond-$ \alpha $ integrals and Montgomery identity

1.
Electrical Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah 80204, Saudi Arabia
2.
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
3.
Lebanese American University, Department of Computer Science and Mathematics, Beirut, Lebanon
4.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt

Received: 25 January 2024 Revised: 13 March 2024 Accepted: 19 March 2024 Published: 02 April 2024
MSC : 39B62, 39B05, 26D15, 26D20, 26D10

In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for $ \diamondsuit_\alpha $-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.
- Gr$ \ddot{u} $ss-type inequality,
- diamond-$ \alpha $ integral,
- Ostrowski inequality,
- trapezoid inequality,
- Montgomery identity
Citation: Marwa M. Tharwat, Marwa M. Ahmed, Ammara Nosheen, Khuram Ali Khan, Iram Shahzadi, Dumitru Baleanu, Ahmed A. El-Deeb. Dynamic inequalities of Grüss, Ostrowski and Trapezoid type via diamond-$ \alpha $ integrals and Montgomery identity[J]. AIMS Mathematics, 2024, 9(5): 12778-12799. doi: 10.3934/math.2024624

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Abstract

In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for $ \diamondsuit_\alpha $-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.

References

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