Point monotonicity and radial convexity of first order approximation mappings

In non-smooth optimization a central role is performed by the concept which replaces the gradient. Several notions of generalized differentiability have been introduced and, within these frameworks, some attempts have been made to characterize non-differentiable functions, by different concepts of generalized monotonicity related to the generalized convexity of the functions. The purpose here is to approach this topic by some kinds of generalized monotonicity and convexity of the first order approximation mappings for the functions, suitable from theoretical and computational points of view.