On generalization of E-convex and composite functions

In this paper the concepts of some generalizations of E-convexity such as logarithmic, and exponential E-convexity are introduced for real-valued functions defined on a Banach space and their relationships with known concepts have been discussed. Further, some applications to optimization of non-differentiable objective functions have been proposed. Also, it is proved that the composition of an affine and a G-convex functional is G-convex, where G stands for convex, quasi-convex, r-quasi convex and m-quasi convex.