Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions

Miguel Vivas-Cortez; Muhammad Aamir Ali; Artion Kashuri; Hüseyin Budak; Miguel Vivas-Cortez; Muhammad Aamir Ali; Artion Kashuri; Hüseyin Budak

doi:10.3934/math.2021546

AIMS Mathematics

2021, Volume 6, Issue 9: 9397-9421. doi: 10.3934/math.2021546

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Research article

Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions

1.
Pontificia Universidad Católica del Ecuador, Facultad de Ciencias Naturales y Exactas, Escuela de Ciencias Físicas y Matemáticas, Sede Quito, Ecuador
2.
Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, 210023, China
3.
Department of Mathematics, Faculty of Technical Science, University "Ismail Qemali", 9400 Vlora, Albania
4.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

Received: 01 March 2021 Accepted: 03 June 2021 Published: 23 June 2021
MSC : 26A33, 26A51, 26D07, 26D10, 26D15

In this work, we establish inequalities of Hermite-Hadamard-Mercer (HHM) type for convex functions by using generalized fractional integrals. The results of our paper are the extensions and refinements of Hermite-Hadamard (HH) and Hermite-Hadamard-Mercer (HHM) type inequalities. We discuss special cases of our main results and give new inequalities of HH and HHM type for different fractional integrals like, Riemann-Liouville (RL) fractional integrals, $ k $-Riemann-Liouville ($ k $-RL) fractional integrals, conformable fractional integrals and fractional integrals of exponential kernel.
- Hermite-Hadamard inequality,
- Hermite-Hadamard-Mercer inequality,
- Hölder inequality,
- Jensen-Mercer inequality,
- convex functions,
- fractional integrals
Citation: Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak. Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions[J]. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546

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Abstract

In this work, we establish inequalities of Hermite-Hadamard-Mercer (HHM) type for convex functions by using generalized fractional integrals. The results of our paper are the extensions and refinements of Hermite-Hadamard (HH) and Hermite-Hadamard-Mercer (HHM) type inequalities. We discuss special cases of our main results and give new inequalities of HH and HHM type for different fractional integrals like, Riemann-Liouville (RL) fractional integrals, $ k $-Riemann-Liouville ($ k $-RL) fractional integrals, conformable fractional integrals and fractional integrals of exponential kernel.

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