Analysis of superquadratic fuzzy interval valued function and its integral inequalities

Dawood Khan; Saad Ihsan Butt; Asfand Fahad; Yuanheng Wang; Bandar Bin Mohsin; Dawood Khan; Saad Ihsan Butt; Asfand Fahad; Yuanheng Wang; Bandar Bin Mohsin

doi:10.3934/math.2025025

AIMS Mathematics

2025, Volume 10, Issue 1: 551-583. doi: 10.3934/math.2025025

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Analysis of superquadratic fuzzy interval valued function and its integral inequalities

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, 54000, Pakistan
2.
School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China
3.
Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
4.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 09 October 2024 Revised: 07 December 2024 Accepted: 20 December 2024 Published: 10 January 2025
MSC : 03E72, 26A33, 26A51, 26D10, 26D15, 26E50

Superquadratic function is a generalization of convex functions. Results based on superquadratic functions are more refined than the results obtained using the notion of convexity. This work aims to provide a new class of superquadratic functions called superquadratic fuzzy-interval-valued function (superquadrtic $ F_{I.V.F} $) and demonstrate its properties using fuzzy order relations. In the space of fuzzy intervals, this relation is also termed as the Kulisch-Miranker order relation defined on such a space level-wise. By leveraging the definition and features of superquadrtic $ F_{I.V.F} $, we come up with improved integral inequalities such as Hermite-Hadamard (H.H) and Jensen type for superquadrtic $ F_{I.V.F} $. Furthermore, we offer fractional representation of inequalities of H.H's types for superquadrtic $ F_{I.V.F} $ with respect to fuzzy interval Riemann-Liouville fractional integral operators. These findings are further validated through specific numerical examples and graphical illustrations, which demonstrate the practical relevance and applicability of the results. We have no doubt that these results will open new avenues for researchers to further explore the notion of superuadraticity.
Citation: Dawood Khan, Saad Ihsan Butt, Asfand Fahad, Yuanheng Wang, Bandar Bin Mohsin. Analysis of superquadratic fuzzy interval valued function and its integral inequalities[J]. AIMS Mathematics, 2025, 10(1): 551-583. doi: 10.3934/math.2025025

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Abstract

Superquadratic function is a generalization of convex functions. Results based on superquadratic functions are more refined than the results obtained using the notion of convexity. This work aims to provide a new class of superquadratic functions called superquadratic fuzzy-interval-valued function (superquadrtic $ F_{I.V.F} $) and demonstrate its properties using fuzzy order relations. In the space of fuzzy intervals, this relation is also termed as the Kulisch-Miranker order relation defined on such a space level-wise. By leveraging the definition and features of superquadrtic $ F_{I.V.F} $, we come up with improved integral inequalities such as Hermite-Hadamard (H.H) and Jensen type for superquadrtic $ F_{I.V.F} $. Furthermore, we offer fractional representation of inequalities of H.H's types for superquadrtic $ F_{I.V.F} $ with respect to fuzzy interval Riemann-Liouville fractional integral operators. These findings are further validated through specific numerical examples and graphical illustrations, which demonstrate the practical relevance and applicability of the results. We have no doubt that these results will open new avenues for researchers to further explore the notion of superuadraticity.

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