Differential Characterization of Quasi-Concave Functions without Twice Differentiability

Abstract

This paper presents a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave, and a sufficient condition for such a function to be strictly quasi-concave. These conditions are applicable to continuously differentiable functions that satisfy a mild additional assumption, and do not require the functions to be twice differentiable. Because this additional assumption is trivially satisfied for twice continuously differentiable functions, our results are pure extensions to classical results.