ANALYSIS ON MULTIPLICATIVE k-ATANGANA–BALEANU FRACTIONAL INTEGRALS WITH APPLICATION TO VARIOUS MERCER-TYPE INEQUALITIES

Yun Long () and Tingsong Du
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Yun Long: Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, P. R. China
Tingsong Du: Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, P. R. China†Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, P. R. China

FRACTALS (fractals), 2025, vol. 33, issue 01, 1-37

Abstract: We innovatively introduce a new group of integral operators called multiplicative k-Atangana–Baleanu fractional integrals. Following the path of this research, we discuss their ∗integrability, continuity, boundedness together with linearity, and successfully construct two Hermite–Hadamard–Mercer inequalities employing this multiplicative fractional integrals, focusing on endpoint- and midpoint-type, respectively. Subsequently, we further propose three identities, involving trapezoid-Mercer-, midpoint-Mercer-, and parameterized-Mercer-type, and combining the function Λ∗ is multiplicatively convex or, for some fixed q > 1, the function (lnΛ∗)q is convex, and we formulate a series of corresponding fractional Mercer-type inequalities. Intended to enhance the readers’ deeper understanding of the results, we offer two instances accompanied by graphical representations, facilitating the validity of the inequalities gained in the current paper. Finally, some applications in multiplicative differential equation, the quadrature formula and special means are presented as well.

Keywords: Multiplicatively Convex Function; Hermite–Hadamard–Mercer-Type Inequalities; Multiplicative k-Atangana–Baleanu Fractional Integrals (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25500033

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