Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications

Jamshed Nasir; Shahid Qaisar; Saad Ihsan Butt; Hassen Aydi; Manuel De la Sen; Jamshed Nasir; Shahid Qaisar; Saad Ihsan Butt; Hassen Aydi; Manuel De la Sen

doi:10.3934/math.2022190

AIMS Mathematics

2022, Volume 7, Issue 3: 3418-3439. doi: 10.3934/math.2022190

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Research article

Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications

1.
Department of Mathematics, COMSATS University Islamabad, Lahore campus, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Sahiwal campus, Pakistan
3.
Institute Supérieur d'Informatique et des Techniques de Communication, Université de sousse, H. Sousse 4000, Tunisia
4.
China Medical University Hospital, China Medical University Taichhung 40402, Taiwan
5.
Institute of Research and development of Processes University of Basque country, 48940, Lejona Spain

Received: 26 September 2021 Accepted: 23 November 2021 Published: 01 December 2021
MSC : Primary: 26A51; Secondary: 26A33, 26D07, 26D10, 26D15

Since the supposed Hermite-Hadamard inequality for a convex function was discussed, its expansions, refinements, and variations, which are called Hermite-Hadamard type inequalities, have been widely explored. The main objective of this article is to acquire new Hermite-Hadamard type inequalities employing the Riemann-Liouville fractional operator for functions whose third derivatives of absolute values are convex and quasi-convex in nature. Some special cases of the newly presented results are discussed as well. As applications, several estimates concerning Bessel functions and special means of real numbers are illustrated.
- convexity,
- quasi-convexity,
- Hermite-Hadamard inequality,
- Hölder's Rogers inequality,
- Young's inequality,
- power mean inequality
Citation: Jamshed Nasir, Shahid Qaisar, Saad Ihsan Butt, Hassen Aydi, Manuel De la Sen. Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications[J]. AIMS Mathematics, 2022, 7(3): 3418-3439. doi: 10.3934/math.2022190

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Abstract

Since the supposed Hermite-Hadamard inequality for a convex function was discussed, its expansions, refinements, and variations, which are called Hermite-Hadamard type inequalities, have been widely explored. The main objective of this article is to acquire new Hermite-Hadamard type inequalities employing the Riemann-Liouville fractional operator for functions whose third derivatives of absolute values are convex and quasi-convex in nature. Some special cases of the newly presented results are discussed as well. As applications, several estimates concerning Bessel functions and special means of real numbers are illustrated.

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