Jensen and Hermite-Hadamard type inclusions for harmonical $ h $-Godunova-Levin functions

Waqar Afzal; Khurram Shabbir; Savin Treanţă; Kamsing Nonlaopon; Waqar Afzal; Khurram Shabbir; Savin Treanţă; Kamsing Nonlaopon

doi:10.3934/math.2023170

AIMS Mathematics

2023, Volume 8, Issue 2: 3303-3321. doi: 10.3934/math.2023170

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Research article

Jensen and Hermite-Hadamard type inclusions for harmonical $ h $-Godunova-Levin functions

1.
Department of Mathemtics, Government College University Lahore (GCUL), Lahore 54000, Pakistan
2.
Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan
3.
Department of Applied Mathematics, University Politehnica of Bucharest, Bucharest 060042, Romania
4.
Academy of Romanian Scientists, 54 Splaiul Independentei, Bucharest 050094, Romania
5.
Fundamental Sciences Applied in Engineering Research Center (SFAI), University Politehnica of Bucharest, Bucharest 060042, Romania
6.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 29 August 2022 Revised: 22 October 2022 Accepted: 02 November 2022 Published: 16 November 2022
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

The role of integral inequalities can be seen in both applied and theoretical mathematics fields. According to the definition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its definitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality, and optimization. In this paper, various types of inequalities are introduced using inclusion relations. The inclusion relation enables us firstly to derive some Hermite-Hadamard inequalities (H.H-inequalities) and then to present Jensen inequality for harmonical $ h $-Godunova-Levin interval-valued functions (GL-IVFS) via Riemann integral operator. Moreover, the findings presented in this study have been verified with the use of useful examples that are not trivial.
- Jensen inequality,
- Hermite-Hadamard inequality,
- Godunova-Levin function,
- harmonic convexity,
- interval valued functions
Citation: Waqar Afzal, Khurram Shabbir, Savin Treanţă, Kamsing Nonlaopon. Jensen and Hermite-Hadamard type inclusions for harmonical $ h $-Godunova-Levin functions[J]. AIMS Mathematics, 2023, 8(2): 3303-3321. doi: 10.3934/math.2023170

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Abstract

The role of integral inequalities can be seen in both applied and theoretical mathematics fields. According to the definition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its definitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality, and optimization. In this paper, various types of inequalities are introduced using inclusion relations. The inclusion relation enables us firstly to derive some Hermite-Hadamard inequalities (H.H-inequalities) and then to present Jensen inequality for harmonical $ h $-Godunova-Levin interval-valued functions (GL-IVFS) via Riemann integral operator. Moreover, the findings presented in this study have been verified with the use of useful examples that are not trivial.

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