On \(t\)-convex functions (Q1766438)

Let \(t \in ]0,1[\). A real-valued function \(f\) defined on an interval \(I \subseteq \mathbb{R}\) is called \(t\)-convex if \(f(tx+(1-t)y) \leq tf(x)+(1-t)f(y)\) for all \(x,y \in I\). The authors show that such functions are characterized by the nonnegativity of their (suitably defined) lower second-order generalized derivatives. From this characterization it follows that \(t\)-convexity is a localizable property, that is, a function is \(t\)-convex if it is so around every point.