Differential Characterization of Quasi-Concave Functions without Twice Differentiability

TL;DR

This paper establishes new necessary and sufficient conditions for quasi-concavity and strict quasi-concavity of functions in Banach spaces, extending classical results to functions without twice differentiability.

Contribution

It provides the first comprehensive criteria for quasi-concavity that do not rely on second derivatives, broadening the scope of analysis for differentiable functions.

Findings

Conditions applicable to continuously differentiable functions

Extension of classical results to non-twice differentiable functions

Applicable in Banach space settings

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.