The definitions of (α,λ)-concave and (G,h)-concave functions, introduced in a previous work, are reconsidered. The first one is simplified by symmetrizing λ. A theorem characterizing (α,λ)-concave differentiable functions is given. Moreover it is shown that it is possible to define a new directional derivative so that the usual properties of classical concave functions still hold. A similar approach is also developed for (G,h)-concave functions: the new directional derivative can be expressed in two equivalent ways and can also be used to define a sort of directional derivative for quasi concave functions intended as a limiting case of (G,h)-concave functions.
