Fractional inequalities of the Hermite–Hadamard type for $ m $-polynomial convex and harmonically convex functions

Eze R. Nwaeze; Muhammad Adil Khan; Ali Ahmadian; Mohammad Nazir Ahmad; Ahmad Kamil Mahmood; Eze R. Nwaeze; Muhammad Adil Khan; Ali Ahmadian; Mohammad Nazir Ahmad; Ahmad Kamil Mahmood

doi:10.3934/math.2021115

AIMS Mathematics

2021, Volume 6, Issue 2: 1889-1904. doi: 10.3934/math.2021115

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Fractional inequalities of the Hermite–Hadamard type for $ m $-polynomial convex and harmonically convex functions

1.
Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA
2.
Department of Mathematics, University of Peshawar, Peshawar, Pakistan
3.
Institute of IR 4.0, The National University of Malaysia, 43600 Bangi, Selangor, Malaysia
4.
High Performance Computing Centre, CISD, Universiti Teknologi Petronas, Seri Iskandar, Perak, Malaysia

Received: 21 October 2020 Accepted: 27 November 2020 Published: 03 December 2020
MSC : 26D15, 26A51, 26D10

In this paper, it is our purpose to establish some new fractional inequalities of the Hermite–Hadamard type for the $ m $-polynomial convex and harmonically convex functions. Our results involve the Caputo–Fabrizio and $ \zeta $-Riemann–Liouville fractional integral operators. They generalize, complement and extend existing results in the literature. By taking $ m\geq 2 $, we deduce loads of new and interesting inequalities. We expect that the thought laid out in this work will provoke advance examinations in this course.
- Hermite–Hadamard,
- $ m $-polynomial convex,
- $ m $-polynomial harmonically convex,
- Riemann–Liouville,
- Caputo–Fabrizio
Citation: Eze R. Nwaeze, Muhammad Adil Khan, Ali Ahmadian, Mohammad Nazir Ahmad, Ahmad Kamil Mahmood. Fractional inequalities of the Hermite–Hadamard type for $ m $-polynomial convex and harmonically convex functions[J]. AIMS Mathematics, 2021, 6(2): 1889-1904. doi: 10.3934/math.2021115

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Abstract

In this paper, it is our purpose to establish some new fractional inequalities of the Hermite–Hadamard type for the $ m $-polynomial convex and harmonically convex functions. Our results involve the Caputo–Fabrizio and $ \zeta $-Riemann–Liouville fractional integral operators. They generalize, complement and extend existing results in the literature. By taking $ m\geq 2 $, we deduce loads of new and interesting inequalities. We expect that the thought laid out in this work will provoke advance examinations in this course.

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