Some new notions of fractional Her mite-Hadamard type in-equalities involving applications to the physical sciences

The term convexity in the frame of fractional calculus is a well-established concept that has assembled significant attention in mathematics and various scientific disciplines for over a century. It offers valuable insights and results in diverse fields, along with practical applications due to its geometric interpretation. Moreover, convexity provides researchers with powerful tools and numerical methods for addressing extensive interconnected problems. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we construct new fractional identities for differentiable preinvex functions to strengthen the recently assigned approach even more. Then utilizing these identities, some generalizations of the Hermite-Hadamard type inequality involving generalized preinvexities in the frame of fractional integral operator, namely Riemann-Liouville (R-L) fractional integrals are explored. Finally, we examined some applications to the q-digamma and Bessel functions via the established results. We used fundamental methods to arrive at our conclusions. We anticipate the techniques and approaches addressed by this study will further pique and spark the researcher's interest.