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Abstract
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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.
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