Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for ( h 1, h 2 )-Convex Functions Pertaining to Total Order Relation

Tareq Saeed, Waqar Afzal, Khurram Shabbir, Savin Treanţă and Manuel De la Sen ()
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Tareq Saeed: Nonlinear Analysis and Applied Mathematics—Research Group, Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Waqar Afzal: Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan
Khurram Shabbir: Department of Mathematics, Government College University Lahore (GCUL), Lahore 54000, Pakistan
Savin Treanţă: Department of Applied Mathematics, University Politehnica of Bucharest, 060042 Bucharest, Romania
Manuel De la Sen: Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), Campus of Leioa, 48940 Leioa, Spain

Mathematics, 2022, vol. 10, issue 24, 1-17

Abstract: There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω = ⟨ ω c , ω r ⟩ = ⟨ ω ¯ + ω ̲ 2 , ω ¯ − ω ̲ 2 ⟩ . A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR- ( h 1 , h 2 ) -convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples.

Keywords: Jensen inequality; ( h 1 , h 2 )-convex function; Hermite–Hadamard inequality; Center-Raius-order relation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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