On (a, b)-convex functions
Copyright policy )On (a, b)-convex functions
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 \(D\).
- Silesian University in Opava Czech Republic
- Medical University of Silesia Poland
convex functions, Convex functions and convex programs in convex geometry, Systems of functional equations and inequalities, Convexity of real functions in one variable, generalizations
convex functions, Convex functions and convex programs in convex geometry, Systems of functional equations and inequalities, Convexity of real functions in one variable, generalizations
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