hasan barsam

In this paper, we explore advanced generalizations of Jensen’s inequality specifically for geometrically-arithmetic (GA)-convex functions. Our primary objective is to extend the classical results associated with convex functions to a broader category, thus contributing to the deeper understanding of GA-convexity. To achieve this,we introduce a new function, denoted by, whichis directly related to a given GA-convex function ϕ. This function  plays a pivotal role in deriving more intricate inequalities, particularly those that involve both Jensen’s inequality and the Jensen- Mercer inequality. These inequalities provide new insights into the behavior of GA-convex functions, offering stronger and more generalized versions of the classical results.Moreover,we demonstrate the utility of these new inequalities by presenting their applications in the context of mean theory. In this setting, the generalized inequalities help to establish novel relationships among various types of means, enriching the existing body of knowledge in this domain. Our findings have the potential to advance research inmathematical inequalities, particularly in the study of convexity and its applications to different areas ofmathematical analysis.