Summary: Starting from the pioneering definition of concavity, many extensions have been suggested in the literature, some of them based on the first and second order approximations. They are mainly linked to practical problems: see, for instance, the case of utility and production functions, risk analysis and, in mathematical programming problems, the ones related to fractional and geometric programming. In our opinion it is very worth looking for the basic idea underlying these properties and a tool may be represented by generalized means as in A. Ben-Tal, J. Optimization Theory Appl. 21, 1-14 (1977; Zbl 0325.26007). In that paper most of the attention has been given to analyze the properties of (G,h)- functions. Our aim is devoted to look for a unifying tool so that most of the known definitions can be considered as a continuous development from the classical case to the quasi-concave one. In this effort we propose a new definition showing that it preserves the most important properties gradually going to more general cases.