Hermite-Hadamard type inclusions via generalized Atangana-Baleanu fractional operator with application

Soubhagya Kumar Sahoo; Fahd Jarad; Bibhakar Kodamasingh; Artion Kashuri; Soubhagya Kumar Sahoo; Fahd Jarad; Bibhakar Kodamasingh; Artion Kashuri

doi:10.3934/math.2022683

AIMS Mathematics

2022, Volume 7, Issue 7: 12303-12321. doi: 10.3934/math.2022683

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Hermite-Hadamard type inclusions via generalized Atangana-Baleanu fractional operator with application

1.
Department of Mathematics, Institute of Technical Education and Research, Siksha 'O' Anusandhan University, Bhubaneswar 751030, India
2.
Department of Mathematics, Çankaya University 06790, Ankara, Turkey
3.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, 9400 Vlora, Albania

Received: 16 March 2022 Revised: 09 April 2022 Accepted: 13 April 2022 Published: 25 April 2022
MSC : 26A33, 26A51, 26D10

Defining new fractional operators and employing them to establish well-known integral inequalities has been the recent trend in the theory of mathematical inequalities. To take a step forward, we present novel versions of Hermite-Hadamard type inequalities for a new fractional operator, which generalizes some well-known fractional integral operators. Moreover, a midpoint type fractional integral identity is derived for differentiable mappings, whose absolute value of the first-order derivatives are convex functions. Moreover, considering this identity as an auxiliary result, several improved inequalities are derived using some fundamental inequalities such as Hölder-İşcan, Jensen and Young inequality. Also, if we take the parameter $ \rho = 1 $ in most of the results, we derive new results for Atangana-Baleanu equivalence. One example related to matrices is also given as an application.
- convex functions,
- Hermite-Hadamard inequality,
- Atangana-Baleanu fractional integral operators,
- Young inequality,
- Jensen's inequality
Citation: Soubhagya Kumar Sahoo, Fahd Jarad, Bibhakar Kodamasingh, Artion Kashuri. Hermite-Hadamard type inclusions via generalized Atangana-Baleanu fractional operator with application[J]. AIMS Mathematics, 2022, 7(7): 12303-12321. doi: 10.3934/math.2022683

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Abstract

Defining new fractional operators and employing them to establish well-known integral inequalities has been the recent trend in the theory of mathematical inequalities. To take a step forward, we present novel versions of Hermite-Hadamard type inequalities for a new fractional operator, which generalizes some well-known fractional integral operators. Moreover, a midpoint type fractional integral identity is derived for differentiable mappings, whose absolute value of the first-order derivatives are convex functions. Moreover, considering this identity as an auxiliary result, several improved inequalities are derived using some fundamental inequalities such as Hölder-İşcan, Jensen and Young inequality. Also, if we take the parameter $ \rho = 1 $ in most of the results, we derive new results for Atangana-Baleanu equivalence. One example related to matrices is also given as an application.

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