Filling the Gap between Lower-C^(1) and Lower-C^(2) Functions

The classes of lower-$C^{1,\alpha}$ functions ($0<\alpha\leq 1$), that is, functions locally representable as a maximum of a compactly parametrized family of continuously differentiable functions with $\alpha$-H\"{o}lder derivative, are hereby introduced. These classes form a strictly decreasing sequence from the larger class of lower-$C^1$ towards the smaller class of lower-$C^2$ 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.