<![CDATA[This article aims to provide a comprehensive review of convex functions, generalizations of convexity concepts, and their applications in optimization and mathematical analysis. Although the basic concepts of convex functions are well understood, the understanding and application of advanced concepts such as quasiconvexity, log-convexity, and conjugate functions remain limited in more complex optimization problems, particularly in the context of statistical parameter estimation and machine learning. This study employs a literature review methodology with a descriptive analysis of theories and concepts developed in books and scientific journals, comparing various approaches in optimization and more complex functional structures. The findings highlight that convex functions possess valuable characteristics for optimization, and generalizations of convexity concepts, such as quasiconvexity and log-convexity, can extend the applications of convex functions in more complex analyses, such as in statistical models and machine learning.]]>
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