This paper investigates a fractional boundary value problem characterized by the \(\digamma\) -Caputo fractional derivative of order \(\mu \in (1,2]\) . By reformulating the problem as an equivalent integral equation, we develop novel generalizations of classical inequalities, including the Lyapunov and Wintner-Hartman inequalities, within the framework of a generalized function \(\digamma\) . These results extend existing mathematical theories and provide a versatile toolset for analyzing complex systems governed by fractional dynamics. We illustrate the utility and applicability of the proposed generalizations through several concrete examples, demonstrating their potential for advancing the understanding of fractional differential equations and their applications in various scientific fields.
