Analytical properties and related inequalities derived from multiplicative Hadamard k-fractional integrals

The present article is intended to address the properties and associated inequalities of multiplicative Hadamard $k$-fractional integrals. The core concept lies in introducing the multiplicative Hadamard $k$-fractional integrals. In this framework, various analytical characteristics they possess, such as ${}^{\ast }$integrability, continuity, commutativity, semigroup property, boundedness, and others, are examined herein. Subsequently, the Hermite–Hadamard-analogous inequalities are formulated for the novelly constructed operators. Meanwhile, an identity is inferred within multiplicative Hadamard $k$-fractional integrals, based on which a series of Bullen-type inequalities are derived in this article, where the function ${\Lambda }^{\ast }$ is GG-convex and the function ${(ln{\Lambda }^{\ast })}^s$ is GA-convex for $s>1$, with a particular focus on discussing the case when $0<s\le 1$. To facilitate a more profound understanding of the outcomes, we offer illustrative examples together with numerical simulations to confirm the consistency of the theoretical results. Finally, applications of the proposed results in multiplicative differential equations, quadrature formulas, and special means for real numbers are investigated as well.