Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 20 (2025), 25 -- 74

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A COMPREHENSIVE REVIEW OF THE PACHPATTE-TYPE INEQUALITY PERTAINING TO FRACTIONAL INTEGRAL OPERATORS

Muhammad Tariq, Asif Ali Shaikh and Sotiris K. Ntouyas

Abstract. In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Pachpatte-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators, such as h-convex, m-convex, s-convex, interval h-convex, harmonically convex, interval harmonically h-convex, LR-convex interval-valued, cr-\mathsfh-convex, n-polynomial harmonically s-type convex, ℑs-convex, (α,m)-convex, p-convex and (p,h)-convex functions. Moreover a variety of classes of preinvex functions are included, such as (s,m,ξ)-preinvex, cr-\mathsfh-preinvex, m-preinvex, n-polynomial preinvex and interval-valued functions. Included in the fractional integral operators are Riemann--Liouville fractional integral, k-Riemann--Liouville fractional integral, (k-r)-Riemann--Liouville fractional integral, Ψ-Riemann--Liouville fractional integral, fractional integral operator of exponential kernel, generalized fractional integral, Caputo-Fabrizio fractional integral, Katugampola fractional integral, conformable fractional integral, genera-lized conformable fractional integral, non-conformable fractional integral and Atangana-Baleanu fractional integral operator.

2020 Mathematics Subject Classification: 26A33; 26A51; 26D07; 26D10; 26D15
Keywords: Pachpatte-type inequalities; convex function; Riemann--Liouville fractional integral; Katugampola fractional integral; Caputo-Fabrizio fractional integral; Atangana-Baleanu fractional integral

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Muhammad Tariq
Department of Basic Sciences and Related Studies,
Mehran University of Engineering and Technology,
Jamshoro 76062, Pakistan
e-mail: captaintariq2187@gmail.com



Asif Ali Shaikh
Department of Basic Sciences and Related Studies,
Mehran University of Engineering and Technology,
Jamshoro 76062, Pakistan
e-mail: asif.shaikh@faculty.muet.edu.pk


Sotiris K. Ntouyas - corresponding author
Department of mathematics,
University of Ioannina,
451 10 Ioannina, Greece
e-mail: sntouyas@uoi.gr

https://www.utgjiu.ro/math/sma