Revisiting the Hermite-Hadamard fractional integral inequality via a Green function

Arshad Iqbal; Muhammad Adil Khan; Noor Mohammad; Eze R. Nwaeze; Yu-Ming Chu; Arshad Iqbal; Muhammad Adil Khan; Noor Mohammad; Eze R. Nwaeze; Yu-Ming Chu

doi:10.3934/math.2020391

AIMS Mathematics

2020, Volume 5, Issue 6: 6087-6107. doi: 10.3934/math.2020391

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Research article

Revisiting the Hermite-Hadamard fractional integral inequality via a Green function

1.
Department of Mathematics, University of Peshawar, Peshawar, Pakistan
2.
Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA
3.
Department of Mathematics, Huzhou University, Huzhou 313000, China
4.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China

Received: 11 May 2020 Accepted: 20 July 2020 Published: 28 July 2020
MSC : 26A51, 26D15, 26E60, 41A55

The Hermite-Hadamard inequality by means of the Riemann-Liouville fractional integral operators is already known in the literature. In this paper, it is our purpose to reconstruct this inequality via a relatively new method called the green function technique. In the process, some identities are established. Using these identities, we obtain loads of new results for functions whose second derivative is convex, monotone and concave in absolute value. We anticipate that the method outlined in this article will stimulate further investigation in this direction.
- Hermite-Hadamard inequality,
- Green function,
- convex function,
- concave function,
- Riemann-Liouville fractional integral
Citation: Arshad Iqbal, Muhammad Adil Khan, Noor Mohammad, Eze R. Nwaeze, Yu-Ming Chu. Revisiting the Hermite-Hadamard fractional integral inequality via a Green function[J]. AIMS Mathematics, 2020, 5(6): 6087-6107. doi: 10.3934/math.2020391

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Abstract

The Hermite-Hadamard inequality by means of the Riemann-Liouville fractional integral operators is already known in the literature. In this paper, it is our purpose to reconstruct this inequality via a relatively new method called the green function technique. In the process, some identities are established. Using these identities, we obtain loads of new results for functions whose second derivative is convex, monotone and concave in absolute value. We anticipate that the method outlined in this article will stimulate further investigation in this direction.

References

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