Advanced Mathematical Models & Applications

This article investigates a novel class of inequalities for s-convex functions in the third sense, utilizing generalized fractional integral operators with respect to another function. We establish two foundational theorems that characterize both left-sided and right-sided fractional integral operators, providing a comprehensive framework for this specific function class. Building upon these core results, we derive a series of theorems that elucidate the intricate relationships between the proposed inequalities. To demonstrate the versatility and scope of our findings, we specialize the results to Riemann-Liouville and Hadamard fractional integral settings. Notably, the theoretical framework is applied to yield significant new inequality relations among key special functions, including Beta and Incomplete Beta functions, Gamma and Incomplete Gamma functions, and Imaginary Error functions, thereby bridging abstract fractional calculus with practical analytic applications. Furthermore, it is demonstrated that one of these inequalities provide significantly sharper bounds than analogue one currently established in the literature.