Hadamard functional integral operators within fractional multiplicative calculus

This research focuses on analyzing multiplicative Hadamard functional fractional integral operators and applies them to establish Bullen-type inequalities for twice ${}^{\ast }$differentiable functions. The investigation begins with the formalization of Hadamard functional integrals in fractional multiplicative calculus, followed by an analysis of their fundamental properties, including continuity, boundedness, ${}^{\ast }$integrability, ${}^{\ast }$linearity, and others. Based on the introduced operators again, we derive a fractional integral identity to establish bounds for Bullen-type inequalities. Our results hold under the assumptions that either (i) the second multiplicative derivative $P^{\ast \ast }$ is multiplicatively convex, or (ii) ${\left(\ln P^{\ast \ast }\right)}^{q_1}$ is convex for $q_1>1$. Additionally, we provide insights into the case $0<q_1\le 1$.