On \((\alpha, a)\)-convex functions
Let \(I\subseteq \mathbb{R}\) be an open interval and \(\alpha, a\in (0,1)\) arbitrarily fixed. Under these conditions we show that there exists a nonconstant function \(f: I\to \mathbb{R}\) such that \[ f(\alpha s+ (1- \alpha) t)\leq af(s)+ (1- a) f(t),\qquad s,t\in I, \] if, and only if, the numbers \(\alpha\) and \(a\) are algebraically conjugate. This gives an answer to a question posed by N. Kuhn. Some corollaries for functions defined on convex and core open subsets of a linear space are also given.
scientific article; zbMATH DE number 734506
scientific article