Hyers–Ulam Stability of 2 D -Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

Waqar Afzal, Daniel Breaz, Mujahid Abbas, Luminiţa-Ioana Cotîrlă, Zareen A. Khan () and Eleonora Rapeanu
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Waqar Afzal: Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan
Daniel Breaz: Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
Mujahid Abbas: Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan
Luminiţa-Ioana Cotîrlă: Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Zareen A. Khan: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Eleonora Rapeanu: Department of Mathematics, “Mircea cel Batran” Naval Academy, 900218 Constanta, Romania

Mathematics, 2024, vol. 12, issue 8, 1-34

Abstract: The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

Keywords: Pachpatte’s inequality; Hermite–Hadamard; Fejer inequality; 2D-convex functions; total order relation; Hyers–Ulam stability; fractional operators (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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