<abstract><p>In this manuscript, we define a special type convex function on Euclidean space and explore it on the Riemannian manifold. We also detail the fundamental properties of special type convex functions and some examples that illustrate the idea. Moreover, to demonstrate the application to the problems of optimization, these special type convex functions are used.</p></abstract>
A class of real functions, which is the generalization of a family of convex functions, is introduced; in this connection, we have defined X-convex, strictly X-convex, quasi-Xconvex, strictly quasi-X-convex, and semi-strictly quasi-X-convex functions. Moreover, in this paper, we give a detailed study of the fundamental properties of these functions with various examples, supporting the concepts. Finally, the study of optimization problems employs quasi-X-convex, semistrictly quasi-X-convex, and strictly quasi-X-convex functions.
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