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.
Eleftherios Gkioulekas email@example.com (2013) On equivalent characterizations of convexity of functions, International Journal of Mathematical Education in Science and Technology, 44:3, 410-417, https://doi.org/10.1080/0020739X.2012.703336
International Journal of Mathematical Education in Science and Technology