Filling the Gap between Lower-C ¹ and Lower-C ² Functions

The classes of lower-

functions (

0<\alpha\leq 1 0 < α ≤ 1

), that is, functions locally representable as a maximum of a compactly parametrized family of continuously differentiable functions with

\alpha α

-Hölder derivative, are hereby introduced. These classes form a strictly decreasing sequence from the larger class of lower-

C^1 C 1

towards the smaller class of lower-

C^2 C 2

functions, and can be analogously characterized via perturbed convex inequalities or via appropriate generalized monotonicity properties of their subdifferentials. Several examples are provided and a complete classification is given.