Some new generalized $ \kappa $–fractional Hermite–Hadamard–Mercer type integral inequalities and their applications

Miguel Vivas-Cortez; Muhammad Uzair Awan; Muhammad Zakria Javed; Artion Kashuri; Muhammad Aslam Noor; Khalida Inayat Noor; Miguel Vivas-Cortez; Muhammad Uzair Awan; Muhammad Zakria Javed; Artion Kashuri; Muhammad Aslam Noor; Khalida Inayat Noor

doi:10.3934/math.2022177

AIMS Mathematics

2022, Volume 7, Issue 2: 3203-3220. doi: 10.3934/math.2022177

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Research article

Some new generalized $ \kappa $–fractional Hermite–Hadamard–Mercer type integral inequalities and their applications

1.
Escuela de Ciencias Físicas y Matemáticas, Facultad de Ciencias Exactas y Naturales Pontificia Universidad Católica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Faculty of Technical Science, University "Ismail Qemali", 9400 Vlora, Albania
4.
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

Received: 01 September 2021 Accepted: 13 October 2021 Published: 26 November 2021
MSC : 26A33, 26A51, 26D07, 26D10, 26D15

In this paper, we have established some new Hermite–Hadamard–Mercer type of inequalities by using $ {\kappa} $–Riemann–Liouville fractional integrals. Moreover, we have derived two new integral identities as auxiliary results. From the applied identities as auxiliary results, we have obtained some new variants of Hermite–Hadamard–Mercer type via $ {\kappa} $–Riemann–Liouville fractional integrals. Several special cases are deduced in detail and some know results are recaptured as well. In order to illustrate the efficiency of our main results, some applications regarding special means of positive real numbers and error estimations for the trapezoidal quadrature formula are provided as well.
- Hermite–Hadamard inequality,
- Jensen–Mercer inequality,
- Hölder's inequality,
- power mean inequality,
- special means,
- error estimation
Citation: Miguel Vivas-Cortez, Muhammad Uzair Awan, Muhammad Zakria Javed, Artion Kashuri, Muhammad Aslam Noor, Khalida Inayat Noor. Some new generalized $ \kappa $–fractional Hermite–Hadamard–Mercer type integral inequalities and their applications[J]. AIMS Mathematics, 2022, 7(2): 3203-3220. doi: 10.3934/math.2022177

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Abstract

In this paper, we have established some new Hermite–Hadamard–Mercer type of inequalities by using $ {\kappa} $–Riemann–Liouville fractional integrals. Moreover, we have derived two new integral identities as auxiliary results. From the applied identities as auxiliary results, we have obtained some new variants of Hermite–Hadamard–Mercer type via $ {\kappa} $–Riemann–Liouville fractional integrals. Several special cases are deduced in detail and some know results are recaptured as well. In order to illustrate the efficiency of our main results, some applications regarding special means of positive real numbers and error estimations for the trapezoidal quadrature formula are provided as well.

References

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