Afzal, WaqarBreaz, DanielAbbas, MujahidCotirla, Luminita-IoanaKhan, Zareen A.Rapeanu, Eleonora2025-02-052025-02-052024-04-19Afzal, W., Breaz, D., Abbas, M., Cotîrlă, L.-I., Khan, Z.A. & Rapeanu, E. Stability of 2D-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. Mathematics 2024, 12, 1238. https://DOI.org/10.3390/math12081238.2227-739010.3390/math12081238http://hdl.handle.net/2263/100542DATA AVAILABILITY STATEMENT : Data used to support the findings are included within the article.The aim of this paper is to introduce a new type of two-dimensional convexity by using totalorder 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.en© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.Pachpatte’s inequalityHermite–HadamardFejer inequality2D-Convex functionsTotal order relationHyers–Ulam stabilityFractional operatorsArticle