Advancements in Convex Analysis Through Inverse Cosine Function with Applications

In this article, we introduce a new class of convex functions called α-inverse cosine convex functions (α-ICCF), which extends the traditional classes. We analyze various algebraic and geometric properties by illustrating the graphs of several significant α-ICCF via visual representations. Utilizing this novel class, we derive the Hermite-Hadamard (HH) inequality and certain refinements for functions whose first derivative in absolute value is α-ICCF. The primary tools employed in deriving the main results include Hölder's inequality, Hölder-Iscan inequality and power-mean integral inequality. Our findings demonstrate that the approximations obtained using Hölder-Iscan and the improved power-mean integral inequality are superior to those derived from other methods. In particular, when α=1, the derived results will coincide with those of classical ICCF. This innovative concept of α-inverse cosine convexity opens new avenues for research, encouraging further exploration of such convexity classes.