A functional inequality characterizing convex functions, conjugacy and a generalization of Hölder's and Minkowski's inequalities (Q752345)

The paper deals with the following functional inequality \(af(s)+bf(t)\leq f(as+bt)\) for \(s,t\in {\mathbb{R}}_+\) and with its reverse \(af(s)+bf(t)\geq f(as+bt)\) for \(s,t\in {\mathbb{R}}_+\), where the parameters a and b satisfy the condition \(0<a<1<a+b\) and f: \({\mathbb{R}}_+\to {\mathbb{R}}_+\). It is proved that the solutions of the first inequality and all solutions of the second one which are bounded in a neighbourhood of 0 and satisfy \(f(0)=0\), have the form \(f(t)=f(1)t\). It is then proved a k-dimensional version of the former result, which gives a generalization of Hölder's and Minkowski's inequalities.