  This study focuses on convex functions and their generalized. Thus, we start
this study by giving the definition of convex functions and some of their
properties and discussing a simple geometric property. Then we generalize
E-convex functions and establish some their properties. Moreover, we give
generalized $ s $-convex functions in the second sense and present some new
inequalities of generalized Hermite-Hadamard type for the class of functions
whose second local fractional derivatives of order $ \alpha $ in absolute value
at certain powers are generalized $ s $-convex functions in the second sense.
At the end, some examples that these inequalities are able to be applied to
some special means are showed.
