Advancements in Convex Analysis Through Inverse Cosine Function with Applications

In this article, we introduce a new class of convex functions called \alpha-inverse cosine convex functions (\alpha-ICCF), which extends the traditional classes. We analyze various algebraic and geometric properties by illustrating the graphs of several significant \alpha-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 \alpha-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 \alpha=۱, the derived results will coincide with those of classical ICCF. This innovative concept of \alpha-inverse cosine convexity opens new avenues for research, encouraging further exploration of such convexity classes.

Keywords:

\alpha-Inverse cosine convex functions , Inverse cosine convex functions , Holders inequality , Holder-Iscan inequality , Improved power-mean integral inequality , Hermite-Hadamard type inequalities