The application of convex function and GA-convex function

A convex function is a function that maps from a convex subset of a vector space to the set of real numbers. Convex functions have some important properties, such as non-negativity, monotonicity, and convexity, which can help us derive and prove inequalities. This paper explores the concepts of convex functions and GA-convex functions, demonstrating their utility in proving a variety of common and complex inequalities. Beginning with an overview of convex functions and their extension to GA-convex functions, the study shows how these mathematical tools can be effectively utilized in the context of inequality proofs. By leveraging the properties of these functions, the paper successfully establishes rigorous proofs for a range of inequalities, highlighting the versatility and applicability of convex and GA-convex functions in mathematical analysis. The properties convex and GA-convex functions allow us to use it to determine the direction of inequalities, prove inequalities, determine the optimal solution of inequalities, and even prove Cauchy inequalities.