On \((a,b)\)-convex functions (Q1193282)

Due to N. Kuhn, a function \(f: D\to R\), where \(D\) is an open convex subset of a real linear space and \(a\), \(b\) belong to (0,1), is called \((a,b)\)-convex if \[ f(ax+(1-a)y)\leq b\cdot f(x)+(1-b)\cdot f(y) \] for all \(x\), \(y\) in \(D\). Main result of the paper is the Theorem. Let \(X\) be a real linear space endowed with a semilinear topology, \(D\) an open, convex set in \(X\), and let \(f: D\to X\) be an \((a,b)\)-convex function. If \(a=b\) and \(a\) or \(b\) is rational, then \(f\) is constant in \(D\).