Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions

Waqar Afzal; Khurram Shabbir; Thongchai Botmart; Waqar Afzal; Khurram Shabbir; Thongchai Botmart

doi:10.3934/math.20221064

AIMS Mathematics

2022, Volume 7, Issue 10: 19372-19387. doi: 10.3934/math.20221064

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Research article

Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions

1.
Department of Mathemtics, Government College University Lahore (GCUL), Lahore 54000, Pakistan
2.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Correction on: AIMS Mathematics 8: 13793-13794.

Received: 16 August 2022 Revised: 23 August 2022 Accepted: 29 August 2022 Published: 01 September 2022
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

Interval analysis distinguishes between inclusion relation and order relation. Under the inclusion relation, convexity and nonconvexity contribute to different kinds of inequalities. The construction and refinement of classical inequalities have received a great deal of attention for many classes of convex as well as nonconvex functions. Convex theory, however, is commonly known to rely on Godunova-Levin functions because their properties enable us to determine inequality terms more precisely than those obtained from convex functions. The purpose of this study was to introduce a ($ \subseteq $) relation to established Jensen-type and Hermite-Hadamard inequalities using $ (h_1, h_2) $-Godunova-Levin interval-valued functions. To strengthen the validity of our results, we provide several examples and obtain some new and previously unknown results.
- Hermite-Hadamard inequality,
- Jensen type inequality,
- interval $ (h_1, h_2) $-Godunova-Levin function
Citation: Waqar Afzal, Khurram Shabbir, Thongchai Botmart. Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions[J]. AIMS Mathematics, 2022, 7(10): 19372-19387. doi: 10.3934/math.20221064

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Abstract

Interval analysis distinguishes between inclusion relation and order relation. Under the inclusion relation, convexity and nonconvexity contribute to different kinds of inequalities. The construction and refinement of classical inequalities have received a great deal of attention for many classes of convex as well as nonconvex functions. Convex theory, however, is commonly known to rely on Godunova-Levin functions because their properties enable us to determine inequality terms more precisely than those obtained from convex functions. The purpose of this study was to introduce a ($ \subseteq $) relation to established Jensen-type and Hermite-Hadamard inequalities using $ (h_1, h_2) $-Godunova-Levin interval-valued functions. To strengthen the validity of our results, we provide several examples and obtain some new and previously unknown results.

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