HYBRID FRACTIONAL INTEGRAL INEQUALITIES IN MULTIPLICATIVE CALCULUS WITH APPLICATIONS

Muhammad Umar (), Saad Ihsan Butt () and Youngsoo Seol
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Muhammad Umar: Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
Saad Ihsan Butt: Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
Youngsoo Seol: Department of Mathematics, Dong-A University, Busan 49315, Korea

FRACTALS (fractals), 2025, vol. 33, issue 01, 1-28

Abstract: Aspects of both hybrid and fractional calculus are combined in the (Proportional Caputo-Hybrid) Pcap operators, which are helpful in solving differential equations with non-integer orders and modeling a variety of complicated phenomena in science and engineering. In this paper, we establish the Pcap operators via multiplicative calculus which are termed as multiplicative Pcap operators. we initially formulate two H.H (Hermite–Hadamard)-type inequalities applicable to multiplicative (geometric) convex function via multiplicative Pcap operators. Subsequently, by leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex function via multiplicative Pcap operators also demonstrating two novel identities applicable to multiplicatively differentiable functions. By leveraging these identities, we then establish inequalities of trapezoid and midpoint types specifically designed for multiplicatively convex functions. Additionally, we explore applications of these findings to special functions and special means.

Keywords: Multiplicative Calculus; Multiplicative Convex Function; Hermite–Hadamard Inequality; Multiplicative Proportional Caputo-Hybrid Operator; Bessel Function (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25500197

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