On equivalent characterizations of convexity of functions

Abstract

A detailed development of the theory of convex functions, not often found in complete form in most textbooks, is given. We adopt the strict secant line definition as the definitive definition of convexity. We then show that for differentiable functions, this definition becomes logically equivalent with the first derivative monotonicity definition and the tangent line definition. Consequently, for differentiable functions, all three characterizations are logically equivalent.

Keywords:

• calculus
• concavity
• convexity

Acknowledgements

It is a pleasure to thank Dr Lokenath Debnath for discussions and for encouraging me to write this article.