Analysis of inclusions using $ s $-type convex interval-valued functions via Riemann-Liouville fractional integrals

Ammara Nosheen; Khuram Ali Khan; Mariam Aslam; Atiq Ur Rehman; Tamador Alihia; Salwa El-Morsy; Ammara Nosheen; Khuram Ali Khan; Mariam Aslam; Atiq Ur Rehman; Tamador Alihia; Salwa El-Morsy

doi:10.3934/math.2026084

AIMS Mathematics

2026, Volume 11, Issue 1: 2027-2045. doi: 10.3934/math.2026084

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Research article

Analysis of inclusions using $ s $-type convex interval-valued functions via Riemann-Liouville fractional integrals

1.
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan
3.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
4.
Basic Science Department, Nile higher institute for engineering and technology, Mansoura, Egypt

Received: 03 November 2025 Revised: 02 January 2026 Accepted: 08 January 2026 Published: 21 January 2026
MSC : 26A33, 26D07, 26E25, 28B20, 65G40

This paper examines a family of convex interval-valued (IVC) functions using the Riemann-Liouville (RL) integrals. Hermite-Hadamard (HH) and Hermite-Hadamard-Fejér (HHF) type inclusions are developed by employing the $ s $-type convexity of interval-valued (Ⅳ) functions. Some inclusions for the product of $ s $-type (IVC) functions are also established involving RL integrals. All main results are furthur refined into inclusions and inequalities for $ s $-type IVC functions and $ s $-type convex point-valued functions, respectively, involving the ordinary integral. In addition, several consequences of the primary results are explored demonstrating the connections between point-valued convex functions, IVC functions, and $ s $-type IVC functions. Each key conclusion is validated numerically. The results of this paper might open the path for new avenues in modeling, optimization problems, interval differential equations, and fuzzy Ⅳ functions that involve both discrete and continuous variables simultaneously.
- convex functions,
- fractional integrals,
- interval-valued functions,
- integral inclusions,
- optimization problems,
- mathematical operators
Citation: Ammara Nosheen, Khuram Ali Khan, Mariam Aslam, Atiq Ur Rehman, Tamador Alihia, Salwa El-Morsy. Analysis of inclusions using $ s $-type convex interval-valued functions via Riemann-Liouville fractional integrals[J]. AIMS Mathematics, 2026, 11(1): 2027-2045. doi: 10.3934/math.2026084

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Abstract

This paper examines a family of convex interval-valued (IVC) functions using the Riemann-Liouville (RL) integrals. Hermite-Hadamard (HH) and Hermite-Hadamard-Fejér (HHF) type inclusions are developed by employing the $ s $-type convexity of interval-valued (Ⅳ) functions. Some inclusions for the product of $ s $-type (IVC) functions are also established involving RL integrals. All main results are furthur refined into inclusions and inequalities for $ s $-type IVC functions and $ s $-type convex point-valued functions, respectively, involving the ordinary integral. In addition, several consequences of the primary results are explored demonstrating the connections between point-valued convex functions, IVC functions, and $ s $-type IVC functions. Each key conclusion is validated numerically. The results of this paper might open the path for new avenues in modeling, optimization problems, interval differential equations, and fuzzy Ⅳ functions that involve both discrete and continuous variables simultaneously.

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