This paper presents the development of reverse Minkowski and reverse Hölder's fractional integral inclusions. We propose a generic class of ${\eta }_{\hslash }$ interval-valued $(\mathrm{I}.\mathrm{V})$ convex functions, which unifies various existing classes. Additionally, we obtain a discrete Jensen-type inclusion within this convexity setup. By leveraging this advanced convexity structure together with tempered fractional integral operators, we derive new Hermite–Hadamard ( $\mathrm{H}$- $\mathrm{H}$)-type, Fejér- $\mathrm{H}$- $\mathrm{H}$-type, and other fractional inclusions. Moreover, we explore the broader significance of our results, supporting them with graphical visualizations. The applications of our results are demonstrated through average value computations.

In this paper, we explored Caputo-Hadamard fractional Wirtinger-type inequalities using Taylor's formula. The main findings were derived by utilizing Hölder's inequality to derive results for Caputo-Hadamard fractional derivatives in terms of $L_q$ norms for $q>1$. Through graphical interpretation, we confirmed the validity of the results. A flowchart summarizing the logical progression from lemma to theorem was added for clarity. Furthermore, inequalities were also derived for Hadamard fractional derivatives. Finally, we discussed the applications of Wirtinger-type inequalitie which incorporates arithmetic mean and geometric mean-type inequalities.

This work introduced normalized representations of generalized Bessel functions, denoted by ${}_k{H}_{\beta ,g}(\mathrm{\wp })$ and ${}_k{M}_{\beta ,g}(\mathrm{\wp };s)$ in terms of $k$ and $(s,k)$, where $s>k>0$ and $s<1$. The motivation stemed from the increasing relevance of such functions in diverse fields such as mathematical physics, wave dynamics and fractional calculus, where classical Bessel functions may be limited. This study also gave detailed geometric investigation of these generalized Bessel functions. We derived sufficient conditions for these functions to be starlike and convex of order $\zeta$ in the open unit disk $\mathbb{U}$. The results of our analysis revealed enhanced structural flexibility over classical counterparts, making them well-suited for advanced applications in complex modeling theory. Several illustrative examples and plots were provided to validate and visualize the behavior of the proposed functions, highlighting their potential in mathematical modeling and theoretical studies. These findings enhanced the geometric function theory framework for special functions and provided a solid foundation for future analytical and applied investigations.