  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-$X$-convex, 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.
